examples of a heuristic problem solving

Heuristics: Definition, Examples, And How They Work

Benjamin Frimodig

Science Expert

B.A., History and Science, Harvard University

Ben Frimodig is a 2021 graduate of Harvard College, where he studied the History of Science.

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Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Every day our brains must process and respond to thousands of problems, both large and small, at a moment’s notice. It might even be overwhelming to consider the sheer volume of complex problems we regularly face in need of a quick solution.

While one might wish there was time to methodically and thoughtfully evaluate the fine details of our everyday tasks, the cognitive demands of daily life often make such processing logistically impossible.

Therefore, the brain must develop reliable shortcuts to keep up with the stimulus-rich environments we inhabit. Psychologists refer to these efficient problem-solving techniques as heuristics.

Heuristics can be thought of as general cognitive frameworks humans rely on regularly to reach a solution quickly.

For example, if a student needs to decide what subject she will study at university, her intuition will likely be drawn toward the path that she envisions as most satisfying, practical, and interesting.

She may also think back on her strengths and weaknesses in secondary school or perhaps even write out a pros and cons list to facilitate her choice.

It’s important to note that these heuristics broadly apply to everyday problems, produce sound solutions, and helps simplify otherwise complicated mental tasks. These are the three defining features of a heuristic.

While the concept of heuristics dates back to Ancient Greece (the term is derived from the Greek word for “to discover”), most of the information known today on the subject comes from prominent twentieth-century social scientists.

Herbert Simon’s study of a notion he called “bounded rationality” focused on decision-making under restrictive cognitive conditions, such as limited time and information.

This concept of optimizing an inherently imperfect analysis frames the contemporary study of heuristics and leads many to credit Simon as a foundational figure in the field.

Kahneman’s Theory of Decision Making

The immense contributions of psychologist Daniel Kahneman to our understanding of cognitive problem-solving deserve special attention.

As context for his theory, Kahneman put forward the estimate that an individual makes around 35,000 decisions each day! To reach these resolutions, the mind relies on either “fast” or “slow” thinking.

The fast thinking pathway (system 1) operates mostly unconsciously and aims to reach reliable decisions with as minimal cognitive strain as possible.

While system 1 relies on broad observations and quick evaluative techniques (heuristics!), system 2 (slow thinking) requires conscious, continuous attention to carefully assess the details of a given problem and logically reach a solution.

Given the sheer volume of daily decisions, it’s no surprise that around 98% of problem-solving uses system 1.

Thus, it is crucial that the human mind develops a toolbox of effective, efficient heuristics to support this fast-thinking pathway.

Heuristics vs. Algorithms

Those who’ve studied the psychology of decision-making might notice similarities between heuristics and algorithms. However, remember that these are two distinct modes of cognition.

Heuristics are methods or strategies which often lead to problem solutions but are not guaranteed to succeed.

They can be distinguished from algorithms, which are methods or procedures that will always produce a solution sooner or later.

An algorithm is a step-by-step procedure that can be reliably used to solve a specific problem. While the concept of an algorithm is most commonly used in reference to technology and mathematics, our brains rely on algorithms every day to resolve issues (Kahneman, 2011).

The important thing to remember is that algorithms are a set of mental instructions unique to specific situations, while heuristics are general rules of thumb that can help the mind process and overcome various obstacles.

For example, if you are thoughtfully reading every line of this article, you are using an algorithm.

On the other hand, if you are quickly skimming each section for important information or perhaps focusing only on sections you don’t already understand, you are using a heuristic!

Why Heuristics Are Used

Heuristics usually occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind at the same moment

When studying heuristics, keep in mind both the benefits and unavoidable drawbacks of their application. The ubiquity of these techniques in human society makes such weaknesses especially worthy of evaluation.

More specifically, in expediting decision-making processes, heuristics also predispose us to a number of cognitive biases .

A cognitive bias is an incorrect but pervasive judgment derived from an illogical pattern of cognition. In simple terms, a cognitive bias occurs when one internalizes a subjective perception as a reliable and objective truth.

Heuristics are reliable but imperfect; In the application of broad decision-making “shortcuts” to guide one’s response to specific situations, occasional errors are both inevitable and have the potential to catalyze persistent mistakes.

For example, consider the risks of faulty applications of the representative heuristic discussed above. While the technique encourages one to assign situations into broad categories based on superficial characteristics and one’s past experiences for the sake of cognitive expediency, such thinking is also the basis of stereotypes and discrimination.

In practice, these errors result in the disproportionate favoring of one group and/or the oppression of other groups within a given society.

Indeed, the most impactful research relating to heuristics often centers on the connection between them and systematic discrimination.

The tradeoff between thoughtful rationality and cognitive efficiency encompasses both the benefits and pitfalls of heuristics and represents a foundational concept in psychological research.

When learning about heuristics, keep in mind their relevance to all areas of human interaction. After all, the study of social psychology is intrinsically interdisciplinary.

Many of the most important studies on heuristics relate to flawed decision-making processes in high-stakes fields like law, medicine, and politics.

Researchers often draw on a distinct set of already established heuristics in their analysis. While dozens of unique heuristics have been observed, brief descriptions of those most central to the field are included below:

Availability Heuristic

The availability heuristic describes the tendency to make choices based on information that comes to mind readily.

For example, children of divorced parents are more likely to have pessimistic views towards marriage as adults.

Of important note, this heuristic can also involve assigning more importance to more recently learned information, largely due to the easier recall of such information.

Representativeness Heuristic

This technique allows one to quickly assign probabilities to and predict the outcome of new scenarios using psychological prototypes derived from past experiences.

For example, juries are less likely to convict individuals who are well-groomed and wearing formal attire (under the assumption that stylish, well-kempt individuals typically do not commit crimes).

This is one of the most studied heuristics by social psychologists for its relevance to the development of stereotypes.

Scarcity Heuristic

This method of decision-making is predicated on the perception of less abundant, rarer items as inherently more valuable than more abundant items.

We rely on the scarcity heuristic when we must make a fast selection with incomplete information. For example, a student deciding between two universities may be drawn toward the option with the lower acceptance rate, assuming that this exclusivity indicates a more desirable experience.

The concept of scarcity is central to behavioral economists’ study of consumer behavior (a field that evaluates economics through the lens of human psychology).

Trial and Error

This is the most basic and perhaps frequently cited heuristic. Trial and error can be used to solve a problem that possesses a discrete number of possible solutions and involves simply attempting each possible option until the correct solution is identified.

For example, if an individual was putting together a jigsaw puzzle, he or she would try multiple pieces until locating a proper fit.

This technique is commonly taught in introductory psychology courses due to its simple representation of the central purpose of heuristics: the use of reliable problem-solving frameworks to reduce cognitive load.

Anchoring and Adjustment Heuristic

Anchoring refers to the tendency to formulate expectations relating to new scenarios relative to an already ingrained piece of information.

Put simply, this anchoring one to form reasonable estimations around uncertainties. For example, if asked to estimate the number of days in a year on Mars, many people would first call to mind the fact the Earth’s year is 365 days (the “anchor”) and adjust accordingly.

This tendency can also help explain the observation that ingrained information often hinders the learning of new information, a concept known as retroactive inhibition.

Familiarity Heuristic

This technique can be used to guide actions in cognitively demanding situations by simply reverting to previous behaviors successfully utilized under similar circumstances.

The familiarity heuristic is most useful in unfamiliar, stressful environments.

For example, a job seeker might recall behavioral standards in other high-stakes situations from her past (perhaps an important presentation at university) to guide her behavior in a job interview.

Many psychologists interpret this technique as a slightly more specific variation of the availability heuristic.

How to Make Better Decisions

Heuristics are ingrained cognitive processes utilized by all humans and can lead to various biases.

Both of these statements are established facts. However, this does not mean that the biases that heuristics produce are unavoidable. As the wide-ranging impacts of such biases on societal institutions have become a popular research topic, psychologists have emphasized techniques for reaching more sound, thoughtful and fair decisions in our daily lives.

Ironically, many of these techniques are themselves heuristics!

To focus on the key details of a given problem, one might create a mental list of explicit goals and values. To clearly identify the impacts of choice, one should imagine its impacts one year in the future and from the perspective of all parties involved.

Most importantly, one must gain a mindful understanding of the problem-solving techniques used by our minds and the common mistakes that result. Mindfulness of these flawed yet persistent pathways allows one to quickly identify and remedy the biases (or otherwise flawed thinking) they tend to create!

Further Information

• Shah, A. K., & Oppenheimer, D. M. (2008). Heuristics made easy: an effort-reduction framework. Psychological bulletin, 134(2), 207.
• Marewski, J. N., & Gigerenzer, G. (2012). Heuristic decision making in medicine. Dialogues in clinical neuroscience, 14(1), 77.
• Del Campo, C., Pauser, S., Steiner, E., & Vetschera, R. (2016). Decision making styles and the use of heuristics in decision making. Journal of Business Economics, 86(4), 389-412.

What is a heuristic in psychology?

A heuristic in psychology is a mental shortcut or rule of thumb that simplifies decision-making and problem-solving. Heuristics often speed up the process of finding a satisfactory solution, but they can also lead to cognitive biases.

Bobadilla-Suarez, S., & Love, B. C. (2017, May 29). Fast or Frugal, but Not Both: Decision Heuristics Under Time Pressure. Journal of Experimental Psychology: Learning, Memory, and Cognition .

Bowes, S. M., Ammirati, R. J., Costello, T. H., Basterfield, C., & Lilienfeld, S. O. (2020). Cognitive biases, heuristics, and logical fallacies in clinical practice: A brief field guide for practicing clinicians and supervisors. Professional Psychology: Research and Practice, 51 (5), 435–445.

Dietrich, C. (2010). “Decision Making: Factors that Influence Decision Making, Heuristics Used, and Decision Outcomes.” Inquiries Journal/Student Pulse, 2(02).

Groenewegen, A. (2021, September 1). Kahneman Fast and slow thinking: System 1 and 2 explained by Sue. SUE Behavioral Design. Retrieved March 26, 2022, from https://suebehaviouraldesign.com/kahneman-fast-slow-thinking/

Kahneman, D., Lovallo, D., & Sibony, O. (2011). Before you make that big decision .

Kahneman, D. (2011). Thinking, fast and slow . Macmillan.

Pratkanis, A. (1989). The cognitive representation of attitudes. In A. R. Pratkanis, S. J. Breckler, & A. G. Greenwald (Eds.), Attitude structure and function (pp. 71–98). Hillsdale, NJ: Erlbaum.

Simon, H.A., 1956. Rational choice and the structure of the environment. Psychological Review .

Tversky, A., & Kahneman, D. (1974). Judgment under Uncertainty: Heuristics and Biases. Science, 185 (4157), 1124–1131.

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Heuristic Problem Solving: A comprehensive guide with 5 Examples

What are heuristics, advantages of using heuristic problem solving, disadvantages of using heuristic problem solving, heuristic problem solving examples, frequently asked questions.

• Speed: Heuristics are designed to find solutions quickly, saving time in problem solving tasks. Rather than spending a lot of time analyzing every possible solution, heuristics help to narrow down the options and focus on the most promising ones.
• Flexibility: Heuristics are not rigid, step-by-step procedures. They allow for flexibility and creativity in problem solving, leading to innovative solutions. They encourage thinking outside the box and can generate unexpected and valuable ideas.
• Simplicity: Heuristics are often easy to understand and apply, making them accessible to anyone regardless of their expertise or background. They don’t require specialized knowledge or training, which means they can be used in various contexts and by different people.
• Cost-effective: Because heuristics are simple and efficient, they can save time, money, and effort in finding solutions. They also don’t require expensive software or equipment, making them a cost-effective approach to problem solving.
• Real-world applicability: Heuristics are often based on practical experience and knowledge, making them relevant to real-world situations. They can help solve complex, messy, or ill-defined problems where other problem solving methods may not be practical.
• Potential for errors: Heuristic problem solving relies on generalizations and assumptions, which may lead to errors or incorrect conclusions. This is especially true if the heuristic is not based on a solid understanding of the problem or the underlying principles.
• Limited scope: Heuristic problem solving may only consider a limited number of potential solutions and may not identify the most optimal or effective solution.
• Lack of creativity: Heuristic problem solving may rely on pre-existing solutions or approaches, limiting creativity and innovation in problem-solving.
• Over-reliance: Heuristic problem solving may lead to over-reliance on a specific approach or heuristic, which can be problematic if the heuristic is flawed or ineffective.
• Lack of transparency: Heuristic problem solving may not be transparent or explainable, as the decision-making process may not be explicitly articulated or understood.
• Trial and error: This heuristic involves trying different solutions to a problem and learning from mistakes until a successful solution is found. A software developer encountering a bug in their code may try other solutions and test each one until they find the one that solves the issue.
• Working backward: This heuristic involves starting at the goal and then figuring out what steps are needed to reach that goal. For example, a project manager may begin by setting a project deadline and then work backward to determine the necessary steps and deadlines for each team member to ensure the project is completed on time.
• Breaking a problem into smaller parts: This heuristic involves breaking down a complex problem into smaller, more manageable pieces that can be tackled individually. For example, an HR manager tasked with implementing a new employee benefits program may break the project into smaller parts, such as researching options, getting quotes from vendors, and communicating the unique benefits to employees.
• Using analogies: This heuristic involves finding similarities between a current problem and a similar problem that has been solved before and using the solution to the previous issue to help solve the current one. For example, a salesperson struggling to close a deal may use an analogy to a successful sales pitch they made to help guide their approach to the current pitch.
• Simplifying the problem: This heuristic involves simplifying a complex problem by ignoring details that are not necessary for solving it. This allows the problem solver to focus on the most critical aspects of the problem. For example, a customer service representative dealing with a complex issue may simplify it by breaking it down into smaller components and addressing them individually rather than simultaneously trying to solve the entire problem.

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What Are Heuristics?

These mental shortcuts lead to fast decisions—and biased thinking

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

Steven Gans, MD is board-certified in psychiatry and is an active supervisor, teacher, and mentor at Massachusetts General Hospital.

Verywell / Cindy Chung 

• History and Origins
• Heuristics vs. Algorithms
• Heuristics and Bias

How to Make Better Decisions

If you need to make a quick decision, there's a good chance you'll rely on a heuristic to come up with a speedy solution. Heuristics are mental shortcuts that allow people to solve problems and make judgments quickly and efficiently. Common types of heuristics rely on availability, representativeness, familiarity, anchoring effects, mood, scarcity, and trial-and-error.

Think of these as mental "rule-of-thumb" strategies that shorten decision-making time. Such shortcuts allow us to function without constantly stopping to think about our next course of action.

However, heuristics have both benefits and drawbacks. These strategies can be handy in many situations but can also lead to  cognitive biases . Becoming aware of this might help you make better and more accurate decisions.

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History of the Research on Heuristics

Nobel-prize winning economist and cognitive psychologist Herbert Simon originally introduced the concept of heuristics in psychology in the 1950s. He suggested that while people strive to make rational choices, human judgment is subject to cognitive limitations. Purely rational decisions would involve weighing every alternative's potential costs and possible benefits.

However, people are limited by the amount of time they have to make a choice and the amount of information they have at their disposal. Other factors, such as overall intelligence and accuracy of perceptions, also influence the decision-making process.

In the 1970s, psychologists Amos Tversky and Daniel Kahneman presented their research on cognitive biases. They proposed that these biases influence how people think and make judgments.

Because of these limitations, we must rely on mental shortcuts to help us make sense of the world.

Simon's research demonstrated that humans were limited in their ability to make rational decisions, but it was Tversky and Kahneman's work that introduced the study of heuristics and the specific ways of thinking that people rely on to simplify the decision-making process.

How Heuristics Are Used

Heuristics play important roles in both  problem-solving  and  decision-making , as we often turn to these mental shortcuts when we need a quick solution.

Here are a few different theories from psychologists about why we rely on heuristics.

• Attribute substitution : People substitute simpler but related questions in place of more complex and difficult questions.
• Effort reduction : People use heuristics as a type of cognitive laziness to reduce the mental effort required to make choices and decisions.
• Fast and frugal : People use heuristics because they can be fast and correct in certain contexts. Some theories argue that heuristics are actually more accurate than they are biased.

In order to cope with the tremendous amount of information we encounter and to speed up the decision-making process, our brains rely on these mental strategies to simplify things so we don't have to spend endless amounts of time analyzing every detail.

You probably make hundreds or even thousands of decisions every day. What should you have for breakfast? What should you wear today? Should you drive or take the bus? Fortunately, heuristics allow you to make such decisions with relative ease and without a great deal of agonizing.

There are many heuristics examples in everyday life. When trying to decide if you should drive or ride the bus to work, for instance, you might remember that there is road construction along the bus route. You realize that this might slow the bus and cause you to be late for work. So you leave earlier and drive to work on an alternate route.

Heuristics allow you to think through the possible outcomes quickly and arrive at a solution.

Are Heuristics Good or Bad?

Heuristics aren't inherently good or bad, but there are pros and cons to using them to make decisions. While they can help us figure out a solution to a problem faster, they can also lead to inaccurate judgments about others or situations. Understanding these pros and cons may help you better use heuristics to make better decisions.

Types of Heuristics

There are many different kinds of heuristics. While each type plays a role in decision-making, they occur during different contexts. Understanding the types can help you better understand which one you are using and when.

Availability

The availability heuristic  involves making decisions based upon how easy it is to bring something to mind. When you are trying to make a decision, you might quickly remember a number of relevant examples.

Since these are more readily available in your memory, you will likely judge these outcomes as being more common or frequently occurring.

For example, imagine you are planning to fly somewhere on vacation. As you are preparing for your trip, you might start to think of a number of recent airline accidents. You might feel like air travel is too dangerous and decide to travel by car instead. Because those examples of air disasters came to mind so easily, the availability heuristic leads you to think that plane crashes are more common than they really are.

Familiarity

The familiarity heuristic refers to how people tend to have more favorable opinions of things, people, or places they've experienced before as opposed to new ones. In fact, given two options, people may choose something they're more familiar with even if the new option provides more benefits.

Representativeness

The representativeness heuristic  involves making a decision by comparing the present situation to the most representative mental prototype. When you are trying to decide if someone is trustworthy, you might compare aspects of the individual to other mental examples you hold.

A soft-spoken older woman might remind you of your grandmother, so you might immediately assume she is kind, gentle, and trustworthy. However, this is an example of a heuristic bias, as you can't know someone trustworthy based on their age alone.

The affect heuristic involves making choices that are influenced by an individual's emotions at that moment. For example, research has shown that people are more likely to see decisions as having benefits and lower risks when in a positive mood.

Negative emotions, on the other hand, lead people to focus on the potential downsides of a decision rather than the possible benefits.

The anchoring bias involves the tendency to be overly influenced by the first bit of information we hear or learn. This can make it more difficult to consider other factors and lead to poor choices. For example, anchoring bias can influence how much you are willing to pay for something, causing you to jump at the first offer without shopping around for a better deal.

Scarcity is a heuristic principle in which we view things that are scarce or less available to us as inherently more valuable. Marketers often use the scarcity heuristic to influence people to buy certain products. This is why you'll often see signs that advertise "limited time only," or that tell you to "get yours while supplies last."

Trial and Error

Trial and error is another type of heuristic in which people use a number of different strategies to solve something until they find what works. Examples of this type of heuristic are evident in everyday life.

People use trial and error when playing video games, finding the fastest driving route to work, or learning to ride a bike (or any new skill).

Difference Between Heuristics and Algorithms

Though the terms are often confused, heuristics and algorithms are two distinct terms in psychology.

Algorithms are step-by-step instructions that lead to predictable, reliable outcomes, whereas heuristics are mental shortcuts that are basically best guesses. Algorithms always lead to accurate outcomes, whereas, heuristics do not.

Examples of algorithms include instructions for how to put together a piece of furniture or a recipe for cooking a certain dish. Health professionals also create algorithms or processes to follow in order to determine what type of treatment to use on a patient.

How Heuristics Can Lead to Bias

Heuristics can certainly help us solve problems and speed up our decision-making process, but that doesn't mean they are always a good thing. They can also introduce errors, bias, and irrational decision-making. As in the examples above, heuristics can lead to inaccurate judgments about how commonly things occur and how representative certain things may be.

Just because something has worked in the past does not mean that it will work again, and relying on a heuristic can make it difficult to see alternative solutions or come up with new ideas.

Heuristics can also contribute to stereotypes and  prejudice . Because people use mental shortcuts to classify and categorize people, they often overlook more relevant information and create stereotyped categorizations that are not in tune with reality.

While heuristics can be a useful tool, there are ways you can improve your decision-making and avoid cognitive bias at the same time.

We are more likely to make an error in judgment if we are trying to make a decision quickly or are under pressure to do so. Taking a little more time to make a decision can help you see things more clearly—and make better choices.

Whenever possible, take a few deep breaths and do something to distract yourself from the decision at hand. When you return to it, you may find a fresh perspective or notice something you didn't before.

Identify the Goal

We tend to focus automatically on what works for us and make decisions that serve our best interest. But take a moment to know what you're trying to achieve. Consider some of the following questions:

• Are there other people who will be affected by this decision?
• What's best for them?
• Is there a common goal that can be achieved that will serve all parties?

Thinking through these questions can help you figure out your goals and the impact that these decisions may have.

Process Your Emotions

Fast decision-making is often influenced by emotions from past experiences that bubble to the surface. Anger, sadness, love, and other powerful feelings can sometimes lead us to decisions we might not otherwise make.

Is your decision based on facts or emotions? While emotions can be helpful, they may affect decisions in a negative way if they prevent us from seeing the full picture.

Recognize All-or-Nothing Thinking

When making a decision, it's a common tendency to believe you have to pick a single, well-defined path, and there's no going back. In reality, this often isn't the case.

Sometimes there are compromises involving two choices, or a third or fourth option that we didn't even think of at first. Try to recognize the nuances and possibilities of all choices involved, instead of using all-or-nothing thinking .

Heuristics are common and often useful. We need this type of decision-making strategy to help reduce cognitive load and speed up many of the small, everyday choices we must make as we live, work, and interact with others.

But it pays to remember that heuristics can also be flawed and lead to irrational choices if we rely too heavily on them. If you are making a big decision, give yourself a little extra time to consider your options and try to consider the situation from someone else's perspective. Thinking things through a bit instead of relying on your mental shortcuts can help ensure you're making the right choice.

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AlKhars M, Evangelopoulos N, Pavur R, Kulkarni S. Cognitive biases resulting from the representativeness heuristic in operations management: an experimental investigation .  Psychol Res Behav Manag . 2019;12:263-276. doi:10.2147/PRBM.S193092

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Teovanović P. Individual differences in anchoring effect: Evidence for the role of insufficient adjustment .  Eur J Psychol . 2019;15(1):8-24. doi:10.5964/ejop.v15i1.1691

Cheung TT, Kroese FM, Fennis BM, De Ridder DT. Put a limit on it: The protective effects of scarcity heuristics when self-control is low . Health Psychol Open . 2015;2(2):2055102915615046. doi:10.1177/2055102915615046

Mohr H, Zwosta K, Markovic D, Bitzer S, Wolfensteller U, Ruge H. Deterministic response strategies in a trial-and-error learning task . Inman C, ed. PLoS Comput Biol. 2018;14(11):e1006621. doi:10.1371/journal.pcbi.1006621

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

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8.2 Problem-Solving: Heuristics and Algorithms

Learning objectives.

• Describe the differences between heuristics and algorithms in information processing.

When faced with a problem to solve, should you go with intuition or with more measured, logical reasoning? Obviously, we use both of these approaches. Some of the decisions we make are rapid, emotional, and automatic. Daniel Kahneman (2011) calls this “fast” thinking. By definition, fast thinking saves time. For example, you may quickly decide to buy something because it is on sale; your fast brain has perceived a bargain, and you go for it quickly. On the other hand, “slow” thinking requires more effort; applying this in the same scenario might cause us not to buy the item because we have reasoned that we don’t really need it, that it is still too expensive, and so on. Using slow and fast thinking does not guarantee good decision-making if they are employed at the wrong time. Sometimes it is not clear which is called for, because many decisions have a level of uncertainty built into them. In this section, we will explore some of the applications of these tendencies to think fast or slow.

We will look further into our thought processes, more specifically, into some of the problem-solving strategies that we use. Heuristics are information-processing strategies that are useful in many cases but may lead to errors when misapplied. A heuristic is a principle with broad application, essentially an educated guess about something. We use heuristics all the time, for example, when deciding what groceries to buy from the supermarket, when looking for a library book, when choosing the best route to drive through town to avoid traffic congestion, and so on. Heuristics can be thought of as aids to decision making; they allow us to reach a solution without a lot of cognitive effort or time.

The benefit of heuristics in helping us reach decisions fairly easily is also the potential downfall: the solution provided by the use of heuristics is not necessarily the best one. Let’s consider some of the most frequently applied, and misapplied, heuristics in the table below.

In many cases, we base our judgments on information that seems to represent, or match, what we expect will happen, while ignoring other potentially more relevant statistical information. When we do so, we are using the representativeness heuristic . Consider, for instance, the data presented in the table below. Let’s say that you went to a hospital, and you checked the records of the babies that were born on that given day. Which pattern of births do you think you are most likely to find?

Most people think that list B is more likely, probably because list B looks more random, and matches — or is “representative of” — our ideas about randomness, but statisticians know that any pattern of four girls and four boys is mathematically equally likely. Whether a boy or girl is born first has no bearing on what sex will be born second; these are independent events, each with a 50:50 chance of being a boy or a girl. The problem is that we have a schema of what randomness should be like, which does not always match what is mathematically the case. Similarly, people who see a flipped coin come up “heads” five times in a row will frequently predict, and perhaps even wager money, that “tails” will be next. This behaviour is known as the gambler’s fallacy . Mathematically, the gambler’s fallacy is an error: the likelihood of any single coin flip being “tails” is always 50%, regardless of how many times it has come up “heads” in the past.

The representativeness heuristic may explain why we judge people on the basis of appearance. Suppose you meet your new next-door neighbour, who drives a loud motorcycle, has many tattoos, wears leather, and has long hair. Later, you try to guess their occupation. What comes to mind most readily? Are they a teacher? Insurance salesman? IT specialist? Librarian? Drug dealer? The representativeness heuristic will lead you to compare your neighbour to the prototypes you have for these occupations and choose the one that they seem to represent the best. Thus, your judgment is affected by how much your neibour seems to resemble each of these groups. Sometimes these judgments are accurate, but they often fail because they do not account for base rates , which is the actual frequency with which these groups exist. In this case, the group with the lowest base rate is probably drug dealer.

Our judgments can also be influenced by how easy it is to retrieve a memory. The tendency to make judgments of the frequency or likelihood that an event occurs on the basis of the ease with which it can be retrieved from memory is known as the availability heuristic (MacLeod & Campbell, 1992; Tversky & Kahneman, 1973). Imagine, for instance, that I asked you to indicate whether there are more words in the English language that begin with the letter “R” or that have the letter “R” as the third letter. You would probably answer this question by trying to think of words that have each of the characteristics, thinking of all the words you know that begin with “R” and all that have “R” in the third position. Because it is much easier to retrieve words by their first letter than by their third, we may incorrectly guess that there are more words that begin with “R,” even though there are in fact more words that have “R” as the third letter.

The availability heuristic may explain why we tend to overestimate the likelihood of crimes or disasters; those that are reported widely in the news are more readily imaginable, and therefore, we tend to overestimate how often they occur. Things that we find easy to imagine, or to remember from watching the news, are estimated to occur frequently. Anything that gets a lot of news coverage is easy to imagine. Availability bias does not just affect our thinking. It can change behaviour. For example, homicides are usually widely reported in the news, leading people to make inaccurate assumptions about the frequency of murder. In Canada, the murder rate has dropped steadily since the 1970s (Statistics Canada, 2018), but this information tends not to be reported, leading people to overestimate the probability of being affected by violent crime. In another example, doctors who recently treated patients suffering from a particular condition were more likely to diagnose the condition in subsequent patients because they overestimated the prevalence of the condition (Poses & Anthony, 1991).

The anchoring and adjustment heuristic is another example of how fast thinking can lead to a decision that might not be optimal. Anchoring and adjustment is easily seen when we are faced with buying something that does not have a fixed price. For example, if you are interested in a used car, and the asking price is $10,000, what price do you think you might offer? Using $10,000 as an anchor, you are likely to adjust your offer from there, and perhaps offer $9000 or $9500. Never mind that $10,000 may not be a reasonable anchoring price. Anchoring and adjustment does not just happen when we’re buying something. It can also be used in any situation that calls for judgment under uncertainty, such as sentencing decisions in criminal cases (Bennett, 2014), and it applies to groups as well as individuals (Rutledge, 1993).

In contrast to heuristics, which can be thought of as problem-solving strategies based on educated guesses, algorithms are problem-solving strategies that use rules. Algorithms are generally a logical set of steps that, if applied correctly, should be accurate. For example, you could make a cake using heuristics — relying on your previous baking experience and guessing at the number and amount of ingredients, baking time, and so on — or using an algorithm. The latter would require a recipe which would provide step-by-step instructions; the recipe is the algorithm. Unless you are an extremely accomplished baker, the algorithm should provide you with a better cake than using heuristics would. While heuristics offer a solution that might be correct, a correctly applied algorithm is guaranteed to provide a correct solution. Of course, not all problems can be solved by algorithms.

As with heuristics, the use of algorithmic processing interacts with behaviour and emotion. Understanding what strategy might provide the best solution requires knowledge and experience. As we will see in the next section, we are prone to a number of cognitive biases that persist despite knowledge and experience.

Key Takeaways

• We use a variety of shortcuts in our information processing, such as the representativeness, availability, and anchoring and adjustment heuristics. These help us to make fast judgments but may lead to errors.
• Algorithms are problem-solving strategies that are based on rules rather than guesses. Algorithms, if applied correctly, are far less likely to result in errors or incorrect solutions than heuristics. Algorithms are based on logic.

Bennett, M. W. (2014). Confronting cognitive ‘anchoring effect’ and ‘blind spot’ biases in federal sentencing: A modest solution for reforming and fundamental flaw. Journal of Criminal Law and Criminology , 104 (3), 489-534.

Kahneman, D. (2011). Thinking, fast and slow. New York, NY: Farrar, Straus and Giroux.

MacLeod, C., & Campbell, L. (1992). Memory accessibility and probability judgments: An experimental evaluation of the availability heuristic.  Journal of Personality and Social Psychology, 63 (6), 890–902.

Poses, R. M., & Anthony, M. (1991). Availability, wishful thinking, and physicians’ diagnostic judgments for patients with suspected bacteremia.  Medical Decision Making,  11 , 159-68.

Rutledge, R. W. (1993). The effects of group decisions and group-shifts on use of the anchoring and adjustment heuristic. Social Behavior and Personality, 21 (3), 215-226.

Statistics Canada. (2018). Ho micide in Canada, 2017 . Retrieved from https://www150.statcan.gc.ca/n1/en/daily-quotidien/181121/dq181121a-eng.pdf

Tversky, A., & Kahneman, D. (1973). Availability: A heuristic for judging frequency and probability.  Cognitive Psychology, 5 , 207–232.

Psychology - 1st Canadian Edition Copyright © 2020 by Sally Walters is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Heuristic Method

Heuristic Method: this article explains the concept of the Heuristic Method , developed by George Pólya in a practical way. After reading it, you will understand the basics of this powerful Problem Solving tool.

What is the Heuristic Method?

A heuristic method is an approach to finding a solution to a problem that originates from the ancient Greek word ‘eurisko’, meaning to ‘find’, ‘search’ or ‘discover’. It is about using a practical method that doesn’t necessarily need to be perfect. Heuristic methods speed up the process of reaching a satisfactory solution.

Previous experiences with comparable problems are used that can concern problem situations for people, machines or abstract issues. One of the founders of heuristics is the Hungarian mathematician György (George) Pólya , who published a book about the subject in 1945 called ‘How to Solve It’. He used four principles that form the basis for problem solving.

Heuristic method: Four principles

Pólya describes the following four principles in his book:

• try to understand the problem
• make a plan
• carry out this plan
• evaluate and adapt

If this sequence doesn’t lead to the right solution, Pólya advises to first look for a simpler problem.

A solution may potentially be found by first looking at a similar problem that was possible to solve. With this experience, it’s possible to look at the current problem in another way.

First principle of the heuristic method: understand the problem

It’s more difficult than it seems, because it seems obvious. In truth, people are hindered when it comes to finding an initially suitable approach to the problem.

It can help to draw the problem and to look at it from another angle. What is the problem, what is happening, can the problem be explained in other words, is there enough information available, etc. Such questions can help with the first evaluation of a problem issue.

Second principle of the heuristic method: make a plan

There are many ways to solve problems. This section is about choosing the right strategy that best fits the problem at hand. The reversed ‘working backwards’ can help with this; people assume to have a solution and use this as a starting point to work towards the problem.

It can also be useful to make an overview of the possibilities, delete some of them immediately, work with comparisons, or to apply symmetry. Creativity comes into play here and will improve the ability to judge.

Third principle of the heuristic method: carry out the plan

Once a strategy has been chosen, the plan can quickly be implemented. However, it is important to pay attention to time and be patient, because the solution will not simply appear.

If the plan doesn’t go anywhere, the advice is to throw it overboard and find a new way.

Fourth principle of the heuristic method: evaluate and adapt

Take the time to carefully consider and reflect upon the work that’s already been done. The things that are going well should be maintained, those leading to a lesser solution, should be adjusted. Some things simply work, while others simply don’t.

There are many different heuristic methods, which Pólya also used. The most well-known heuristics are found below:

1. Dividing technique

The original problem is divided into smaller sub-problems that can be solved more easily. These sub-problems can be linked to each other and combined, which will eventually lead to the solving of the original problem.

2. Inductive method

This involves a problem that has already been solved, but is smaller than the original problem. Generalisation can be derived from the previously solved problem, which can help in solving the bigger, original problem.

3. Reduction method

Because problems are often larger than assumed and deal with different causes and factors, this method sets limits for the problem in advance. This reduces the leeway of the original problem, making it easier to solve.

4. Constructive method

This is about working on the problem step by step. The smallest solution is seen as a victory and from that point consecutive steps are taken. This way, the best choices keep being made, which will eventually lead to a successful end result.

5. Local search method

This is about the search for the most attainable solution to the problem. This solution is improved along the way. This method ends when improvement is no longer possible.

Exact solutions versus the heuristic method

The heuristic approach is a mathmatical method with which proof of a good solution to a problem is delivered. There is a large number of different problems that could use good solutions. When the processing speed is equally as important as the obtained solution, we speak of a heuristic method.

The Heuristic Method only tries to find a good, but not necessarily optimal, solution. This is what differentiates heuristics from exact solution methods, which are about finding the optimal solution to a problem. However, that’s very time consuming, which is why a heuristic method may prove preferable. This is much quicker and more flexible than an exact method, but does have to satisfy a number of criteria.

It’s Your Turn

What do you think? Is the Heuristic Method applicable in your personal or professional environment? Do you recognize the practical explanation or do you have more suggestions? What are your success factors for solving problems

Share your experience and knowledge in the comments box below.

More information

• Groner, R., Groner, M., & Bischof, W. F. (2014). Methods of heuristics . Routledge .
• Newell, A. (1983). The heuristic of George Polya and its relation to artificial intelligence . Methods of heuristics, 195-243.
• Polya, G. (2014, 1945). How to solve it: A new aspect of mathematical method . Princeton university press .

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AP®︎/College Computer Science Principles

Course: ap®︎/college computer science principles   >   unit 4, using heuristics.

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7.3 Problem Solving

Learning objectives.

By the end of this section, you will be able to:

• Describe problem solving strategies
• Define algorithm and heuristic
• Explain some common roadblocks to effective problem solving and decision making

People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

Problem-Solving Strategies

When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them ( Table 7.2 ). For example, a well-known strategy is trial and error . The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Everyday Connection

Solving puzzles.

Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below ( Figure 7.7 ) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

Here is another popular type of puzzle ( Figure 7.8 ) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Take a look at the “Puzzling Scales” logic puzzle below ( Figure 7.9 ). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

Pitfalls to Problem Solving

Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but they just need to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. Duncker (1945) conducted foundational research on functional fixedness. He created an experiment in which participants were given a candle, a book of matches, and a box of thumbtacks. They were instructed to use those items to attach the candle to the wall so that it did not drip wax onto the table below. Participants had to use functional fixedness to overcome the problem ( Figure 7.10 ). During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

Link to Learning

Check out this Apollo 13 scene about NASA engineers overcoming functional fixedness to learn more.

Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in Table 7.3 .

Watch this teacher-made music video about cognitive biases to learn more.

Were you able to determine how many marbles are needed to balance the scales in Figure 7.9 ? You need nine. Were you able to solve the problems in Figure 7.7 and Figure 7.8 ? Here are the answers ( Figure 7.11 ).

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• What are heuristics and how do they hel ...

What are heuristics and how do they help us make decisions?

Heuristics are simple rules of thumb that our brains use to make decisions. When you choose a work outfit that looks professional instead of sweatpants, you’re making a decision based on past information. That's not intuition; it’s heuristics. Instead of weighing all the information available to make a data-backed choice, heuristics enable us to move quickly into action—mostly without us even realizing it. In this article, you’ll learn what heuristics are, their common types, and how we use them in different scenarios.

Green means go. Most of us accept this as common knowledge, but it’s actually an example of a micro-decision—in this case, your brain is deciding to go when you see the color green.

You make countless of these subconscious decisions every day. Many things that you might think just come naturally to you are actually caused by heuristics—mental shortcuts that allow you to quickly process information and take action. Heuristics help you make smaller, almost unnoticeable decisions using past information, without much rational input from your brain.

Heuristics are helpful for getting things done more quickly, but they can also lead to biases and irrational choices if you’re not aware of them. Luckily, you can use heuristics to your advantage once you recognize them, and make better decisions in the workplace.

What is a heuristic?

Heuristics are mental shortcuts that your brain uses to make decisions. When we make rational choices, our brains weigh all the information, pros and cons, and any relevant data. But it’s not possible to do this for every single decision we make on a day-to-day basis. For the smaller ones, your brain uses heuristics to infer information and take almost-immediate action.

Decision-making tools for agile businesses

In this ebook, learn how to equip employees to make better decisions—so your business can pivot, adapt, and tackle challenges more effectively than your competition.

How heuristics work

For example, if you’re making a larger decision about whether to accept a new job or stay with your current one, your brain will process this information slowly. For decisions like this, you collect data by referencing sources—chatting with mentors, reading company reviews, and comparing salaries. Then, you use that information to make your decision. Meanwhile, your brain is also using heuristics to help you speed along that track. In this example, you might use something called the “availability heuristic” to reference things you’ve recently seen about the new job. The availability heuristic makes it more likely that you’ll remember a news story about the company’s higher stock prices. Without realizing it, this can make you think the new job will be more lucrative.

On the flip side, you can recognize that the new job has had some great press recently, but that might be just a great PR team at work. Instead of “buying in” to what the availability heuristic is trying to tell you—that positive news means it’s the right job—you can acknowledge that this is a bias at work. In this case, comparing compensation and work-life balance between the two companies is a much more effective way to choose which job is right for you.

History of heuristics

The term "heuristics," originating from the Greek word meaning “to discover,” has ancient roots, but much of today's understanding comes from twentieth-century social scientists. Herbert Simon's research into "bounded rationality" highlighted the use of heuristics in decision-making, particularly under constraints like limited time and information.

Daniel Kahneman was one of the first researchers to study heuristics in his behavioral economics work in the 1970’s, along with fellow psychologist Amos Tversky. They theorized that many of the decisions and judgments we make aren’t rational—meaning we don’t move through a series of decision-making steps to come to a solution. Instead, the human brain uses mental shortcuts to form seemingly irrational, “fast and frugal” decisions—quick choices that don’t require a lot of mental energy.

Kahneman’s work showed that heuristics lead to systematic errors (or biases), which act as the driving force for our decisions. He was able to apply this research to economic theory, leading to the formation of behavioral economics and a Nobel Prize for Kahneman in 2002.

In the years since, the study of heuristics has grown in popularity with economists and in cognitive psychology. Gerd Gigerenzer’s research , for example, challenges the idea that heuristics lead to errors or flawed thinking. He argues that heuristics are actually indicators that human beings are able to make decisions more effectively without following the traditional rules of logic. His research seems to indicate that heuristics lead us to the right answer most of the time.

Types of heuristics

Heuristics are everywhere, whether we notice them or not. There are hundreds of heuristics at play in the human brain, and they interact with one another constantly. To understand how these heuristics can help you, start by learning some of the more common types of heuristics.

Recognition heuristic

The recognition heuristic uses what we already know (or recognize) as a criterion for decisions. The concept is simple: When faced with two choices, you’re more likely to choose the item you recognize versus the one you don’t.

This is the very base-level concept behind branding your business, and we see it in all well-known companies. Businesses develop a brand messaging strategy in the hopes that when you’re faced with buying their product or buying someone else's, you recognize their product, have a positive association with it, and choose that one. For example, if you’re going to grab a soda and there are two different cans in the fridge, one a Coca-Cola, and the other a soda you’ve never heard of, you are more likely to choose the Coca-Cola simply because you know the name.

Familiarity heuristic

The familiarity heuristic is a mental shortcut where individuals prefer options or information that is familiar to them. This heuristic is based on the notion that familiar items are seen as safer or superior. It differs from the recognition heuristic, which relies solely on whether an item is recognized. The familiarity heuristic involves a deeper sense of comfort and understanding, as opposed to just recognizing something.

An example of this heuristic is seen in investment decisions. Investors might favor well-known companies over lesser-known ones, influenced more by brand familiarity than by an objective assessment of the investment's potential. This tendency showcases how the familiarity heuristic can lead to suboptimal choices, as it prioritizes comfort and recognition over a thorough evaluation of all available options.

Availability heuristic

The availability heuristic is a cognitive bias where people judge the frequency or likelihood of events based on how easily similar instances come to mind. This mental shortcut depends on the most immediate examples that pop into one's mind when considering a topic or decision. The ease of recalling these instances often leads to a distorted perception of their actual frequency, as recent, dramatic, or emotionally charged memories tend to be more memorable.

A notable example of the availability heuristic is the public's reaction to shark attacks. When the media reports on shark attacks, these incidents become highly memorable due to their dramatic nature, leading people to overestimate the risk of such events. This heightened perception is despite statistical evidence showing the rarity of shark attacks. The result is an exaggerated fear and a skewed perception of the actual danger of swimming in the ocean.

Representativeness heuristic

The representativeness heuristic is when we try to assign an object to a specific category or idea based on past experiences. Oftentimes, this comes up when we meet people—our first impression. We expect certain things (such as clothing and credentials) to indicate that a person behaves or lives a certain way.

Without proper awareness, this heuristic can lead to discrimination in the workplace. For example, representativeness heuristics might lead us to believe that a job candidate from an Ivy League school is more qualified than one from a state university, even if their qualifications show us otherwise. This is because we expect Ivy League graduates to act a certain way, such as by being more hard-working or intelligent. Of course, in our rational brains, we know this isn’t the case. That’s why it’s important to be aware of this heuristic, so you can use logical thinking to combat potential biases.

Anchoring and adjustment heuristic

Used in finance for economic forecasting, anchoring and adjustment is when you start with an initial piece of information (the anchor) and continue adjusting until you reach an acceptable decision. The challenge is that sometimes the anchor ends up not being a good enough value to begin with. In other words, you choose the anchor based on unknown biases and then make further decisions based on this faulty assumption.

Anchoring and adjustment are often used in pricing, especially with SaaS companies. For example, a displayed, three-tiered pricing model shows you how much you get for each price point. The layout is designed to make it look like you won’t get much for the lower price, and you don’t necessarily need the highest price, so you choose the mid-level option (the original target). The anchors are the low price (suggesting there’s not much value here) and the high price (which shows that you’re getting a "discount" if you choose another option). Thanks to those two anchors, you feel like you’re getting a lot of value, no matter what you spend.

Affect heuristic

You know the advice; think with your heart. That’s the affect heuristic in action, where you make a decision based on what you’re feeling. Emotions are important ways to understand the world around us, but using them to make decisions is irrational and can impact your work.

For example, let’s say you’re about to ask your boss for a promotion. As a product marketer, you’ve made a huge impact on the company by helping to build a community of enthusiastic, loyal customers. But the day before you have your performance review , you find out that a small project you led for a new product feature failed. You decide to skip the conversation asking for a raise and instead double down on how you can improve.

In this example, you’re using the affect heuristic to base your entire performance on the failure of one small project—even though the rest of your performance (building that profitable community) is much more impactful than a new product feature. If you weighed the options rationally, you would see that asking for a raise is still a logical choice. But instead, the fear of asking for a raise after a failure felt like too big a trade-off.

Satisficing

Satisficing is when you accept an available option that’s satisfactory (i.e., just fine) instead of trying to find the best possible solution. In other words, you’re settling. This creates a “bounded rationality,” where you’re constrained by the choices that are good-enough, instead of pushing past the limits to discover more. This isn’t always negative—for lower-impact scenarios, it might not make sense to invest time and energy into finding the optimal choice. But there are also times when this heuristic kicks in and you end up settling for less than what’s possible.

For example, let’s say you’re a project manager planning the budget for the next fiscal year. Instead of looking at previous spend and revenue, you satisfice and base the budget off projections, assuming that will be good enough. But without factoring in historical data, your budget isn’t going to be as equipped to manage hiccups or unexpected changes. In this case, you can mitigate satisficing with a logically-based data review that, while longer, will produce a more accurate and thoughtful budget plan.

Trial and error heuristic

The trial and error heuristic is a problem-solving method where solutions are found through repeated experimentation. It's used when a clear path to the solution isn't known, relying on iterative learning from failures and adjustments.

For example, a chef might experiment with various ingredient combinations and techniques to perfect a new recipe. Each attempt informs the next, demonstrating how trial and error facilitates discovery in situations without formal guidelines.

Pros and cons of heuristics

Heuristics are effective at helping you get more done quickly, but they also have downsides. Psychologists don’t necessarily agree on whether heuristics and biases are positive or negative. But the argument seems to boil down to these two pros and cons:

Heuristics pros:

Simple heuristics reduce cognitive load, allowing you to accomplish more in less time with fast and frugal decisions. For example, the satisficing heuristic helps you find a "good enough" choice. So if you’re making a complex decision between whether to cut costs or invest in employee well-being , you can use satisficing to find a solution that’s a compromise. The result might not be perfect, but it allows you to take action and get started—you can always adjust later on.

Heuristics cons:

Heuristics create biases. While these cognitive biases enable us to make rapid-fire decisions, they can also lead to rigid, unhelpful beliefs. For example, confirmation bias makes it more likely that you’ll seek out other opinions that agree with your own. This makes it harder to keep an open mind, hear from the other side, and ultimately change your mind—which doesn’t help you build the flexibility and adaptability so important for succeeding in the workplace.

Heuristics and psychology

Heuristics play a pivotal role in psychology, especially in understanding how people make decisions within their cognitive limitations. These mental shortcuts allow for quicker decisions, often necessary in a fast-paced world, but they can sometimes lead to errors in judgment.

The study of heuristics bridges various aspects of psychology, from cognitive processes to behavioral outcomes, and highlights the balance between efficient decision-making and the potential for bias.

Stereotypes and heuristic thinking

Stereotypes are a form of heuristic where individuals make assumptions based on group characteristics, a process analyzed in both English and American psychology.

While these generalizations can lead to rapid conclusions and rational decisions under certain circumstances, they can also oversimplify complex human behaviors and contribute to prejudiced attitudes. Understanding stereotypes as a heuristic offers insight into the cognitive limitations of the human mind and their impact on social perceptions and interactions.

How heuristics lead to bias

Because heuristics rely on shortcuts and stereotypes, they can often lead to bias. This is especially true in scenarios where cognitive limitations restrict the processing of all relevant information. So how do you combat bias? If you acknowledge your biases, you can usually undo them and maybe even use them to your advantage. There are ways you can hack heuristics, so that they work for you (not against you):

Be aware. Heuristics often operate like a knee-jerk reaction—they’re automatic. The more aware you are, the more you can identify and acknowledge the heuristic at play. From there, you can decide if it’s useful for the current situation, or if a logical decision-making process is best.

Flip the script. When you notice a negative bias, turn it around. For example, confirmation bias is when we look for things to be as we expect. So if we expect our boss to assign us more work than our colleagues, we might always experience our work tasks as unfair. Instead, turn this around by repeating that your boss has your team’s best interests at heart, and you know everyone is working hard. This will re-train your confirmation bias to look for all the ways that your boss is treating you just like everyone else.

Practice mindfulness. Mindfulness helps to build self-awareness, so you know when heuristics are impacting your decisions. For example, when we tap into the empathy gap heuristic, we’re unable to empathize with someone else or a specific situation. However, if we’re mindful, we can be aware of how we’re feeling before we engage. This helps us to see that the judgment stems from our own emotions and probably has nothing to do with the other person.

Examples of heuristics in business

This is all well and good in theory, but how do heuristic decision-making and thought processes show up in the real world? One reason researchers have invested so much time and energy into learning about heuristics is so that they can use them, like in these scenarios:

How heuristics are used in marketing

Effective marketing does so much for a business—it attracts new customers, makes a brand a household name, and converts interest into sales, to name a few. One way marketing teams are able to accomplish all this is by applying heuristics.

Let’s use ambiguity aversion as an example. Ambiguity aversion means you're less likely to choose an item you don’t know. Marketing teams combat this by working to become familiar to their customers. This could include the social media team engaging in a more empathetic or conversational way, or employing technology like chat-bots to show that there’s always someone available to help. Making the business feel more approachable helps the customer feel like they know the brand personally—which lessens ambiguity aversion.

How heuristics are used in business strategy

Have you ever noticed how your CEO seems to know things before they happen? Or that the CFO listens more than they speak? These are indications that they understand people in a deeper way, and are able to engage with their employees and predict outcomes because of it. C-suite level executives are often experts in behavioral science, even if they didn’t study it. They tend to get what makes people tick, and know how to communicate based on these biases. In short, they use heuristics for higher-level decision-making processes and execution. 

This includes business strategy . For example, a startup CEO might be aware of their representativeness bias towards investors—they always look for the person in the room with the  fancy suit or car. But after years in the field, they know logically that this isn’t always true—plenty of their investors have shown up in shorts and sandals. Now, because they’re aware of their bias, they can build it into their investment strategy. Instead of only attending expensive, luxury events, they also attend conferences with like-minded individuals and network among peers. This approach can lead them to a greater variety of investors and more potential opportunities.

Heuristics vs algorithms

Heuristics and algorithms are both used by the brain to reduce the mental effort of decision-making, but they operate a bit differently. Algorithms act as guidelines for specific scenarios. They have a structured process designed to solve that specific problem. Heuristics, on the other hand, are general rules of thumb that help the brain process information and may or may not reach a solution.

For example, let's say you’re cooking a well-loved family recipe. You know the steps inside and out, and you no longer need to reference the instructions. If you’re following a recipe step-by-step, you’re using an algorithm. If, however, you decide on a whim to sub in some of your fresh garden vegetables because you think it will taste better, you’re using a heuristic.

How to use heuristics to make better decisions

Heuristics can help us make decisions quickly and with less cognitive strain. While they can be efficient, they sometimes lead to errors in judgment. Understanding how to use heuristics effectively can improve decision-making, especially in complex or uncertain situations.

Take time to think

Rushing often leads to reliance on automatic heuristics, which might not always be suitable. To make better decisions, slow down your thinking process. Take a step back, breathe, and allow yourself a moment of distraction. This pause can provide a fresh perspective and help you notice details or angles you might have missed initially.

Clarify your objectives

When making a decision, it's important to understand the ultimate goal. Our automatic decision-making processes tend to favor immediate benefits, sometimes overlooking long-term impacts or the needs of others involved. Consider the broader implications of your decision. Who else is affected? Is there a common objective that benefits all parties? Such considerations can lead to more holistic and effective decisions.

Manage your emotional influences

Emotions significantly influence our decision-making, often without our awareness. Fast decisions are particularly prone to emotional biases. Acknowledge your feelings, but also separate them from the facts at hand. Are you making a decision based on solid information or emotional reactions? Distinguishing between the two can lead to more rational and balanced choices.

Beware of binary thinking

All-or-nothing thinking is a common heuristic trap, where we see decisions as black or white with no middle ground. However, real-life decisions often have multiple paths and possibilities. It's important to recognize this complexity. There might be compromises or alternative options that weren't initially considered. By acknowledging the spectrum of possibilities, you can make more nuanced and effective decisions.

Heuristic FAQs

What is heuristic thinking.

Heuristic thinking refers to a method of problem-solving, learning, or discovery that employs a practical approach—often termed a "rule of thumb"—to make decisions quickly. Heuristic thinking is a type of cognition that humans use subconsciously to make decisions and judgments with limited time.

What is a heuristic evaluation?

A heuristic evaluation is a usability inspection method used in the fields of user interface (UI) and user experience (UX) design. It involves evaluators examining the interface and judging its compliance with recognized usability principles, known as heuristics. These heuristics serve as guidelines to identify usability problems in a design, making the evaluation process more systematic and comprehensive.

What are computer heuristics?

Computer heuristics are algorithms used to solve complex problems or make decisions where an exhaustive search is impractical. In fields like artificial intelligence and cybersecurity, these heuristic methods allow for efficient problem-solving and decision-making, often based on trial and error or rule-of-thumb strategies.

What are heuristics in psychology?

In psychology, heuristics are quick mental rules for making decisions. They are important in social psychology for understanding how we think and decide. Figures like Kahneman and Tversky, particularly in their work "Judgment Under Uncertainty: Heuristics and Biases," have influenced the study of heuristics in psychology.

Learn heuristics, de-mystify your brain

Your brain doesn’t actually work in mysterious ways. In reality, researchers know why we do a lot of the things we do. Heuristics help us to understand the choices we make that don’t make much sense. Once you understand heuristics, you can also learn to use them to your advantage—both in business, and in life. 

Related resources

How Asana streamlines strategic planning with work management

How to create a CRM strategy: 6 steps (with examples)

What is management by objectives (MBO)?

Write better AI prompts: A 4-sentence framework

22 Heuristics Examples (The Types of Heuristics)

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Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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A heuristic is a mental shortcut that enables people to make quick but less-than-optimal decisions.

The benefit of heuristics is that they allow us to make fast decisions based upon approximations, fast cognitive strategies, and educated guesses. The downside is that they often lead us to come to inaccurate conclusions and make flawed decisions.

The most common examples of heuristics are the availability, representativeness, and affect heuristics. However, there are many more possible examples, as shown in the 23 listed below.

Heuristics Definition

Psychologists Amos Tversky and Daniel Kahneman created the concept of heuristics in the early 1970s. They can be described in the following way:

“[They are] judgmental shortcuts that generally get us where we need to go – and quickly – but at the cost of occasionally sending us off course.”

Thus, we can see heuristics as being both positive and negative for our lives. But most interestingly, they can be leveraged in marketing situations to manipulate people’s purchasing decisions, as discussed below.

Types of Heuristics with Examples

1. availability heuristic.

Quick Definition: Making decisions based upon information that is easily available.

We often rely upon and place greater emphasis upon information that is easily available when making decisions.

We might make a decision based solely on what we know about a topic rather than conducting deeper research in order to make a more informed decision. This causes mistakes in our thinking and leads us to make decisions that are flawed or not sufficiently thought out.

This bias is one reason why political parties try to be the last person who talks to a voter before they go into a polling booth. The newness of the information may cause someone to vote for that part because the party’s arguments are closest to the top of mind.

> Check out these 15 availability heuristic examples

2. Representativeness Heuristic

Quick Definition: Making judgments based upon the similarity of one thing to its archetype. In social situations, this leads to prejudice.

We often make a snap judgment about something by placing it into a category based on its surface appearance. For example, we might see a tree and immediately assume it’s in the oak family based upon the color of its bark or size of its leaves.

In social sciences, we can also see that people make judgements about other people based upon their race, gender, class, or other aspects of their identity. In these situations, we are using stereotypes to come to snap judgements about others.

In these situations, our stereotypical assumptions about others can lead to bias, prejudice , and even discrimination .

> Check out these 11 representativeness heuristic examples

3. Affect Heuristic

Quick Definition: We often make decisions based on emotions, moods, and “gut feelings” rather than logic.

Emotions, moods, and feelings impact our thoughts. This simple fact can lead people into making emotional decisions that they may regret later on when they reflect using logic.

One affect heuristic example is the fact that we often make emotional outbursts that we regret later on. Yelling at a cashier at the shops, for example, may be followed up with regret when we reflect and realize it really wasn’t the cashier’s fault.

Similarly, shoppers make impulse purchases based on the feelings they have about the handbag or new dress. These purchases may be regretted later on when we use logic and realize we have overspent our budgets.

4. Anchoring Heuristic

Quick Definition: We often make decisions based upon a subjective anchoring point that influences all subsequent thinking on a topic.

An anchoring point is often the original piece of information that we are given. Based upon this original piece of information, all future thinking and decisions look good or bad.

An anchoring heuristic example is when a company sets the cost of their goods high before setting a discount. If a high price is set, then a discount is applied, then people would see the price as a bargain rather than high .

Similarly, if you were looking at two highly-priced products, the product that is a few dollars less than the other is seen as a good deal, even if its price is also inflated.

5. Base Rate Heuristic

Quick Definition: We neglect the base statistics in favor of other more proximate statistics when making a judgment.

Base rate neglect occurs when someone forgets the base rate, or a basic fact about information, and instead makes decisions based upon other information that they place too much importance upon.

For example, we may predict that the next person to walk into a hospital is a man if the last three people who entered were all males.

This assumption neglects the fact that 50% of all people who enter hospitals are women.

Here, we are privileging immediate information: that there appears to be a lot of men entering the hospital right now., instead of the base rate fact: that you’ve generally got a 50% chance of a woman walking into the store.

6. Absurdity Heuristic

Quick Definition: We tend to classify things that are improbably as absurd rather than giving them proper consideration.

Many people who believe themselves to be highly logical fall prey to the absurdity heuristic. This occurs when you hear a claim that is improbable, so you instantly dismiss it out of hand.

The ability to filter out absurdity has been highly useful to humans – allowing us to keep our focus on reality and not get caught up in conspiracy theories day and night.

But this becomes a problem when we dismiss things that are serious problems. For example, rejection of climate change science based on the fact that it seems extreme, or a doctor dismissing symptoms of a rare disease, are cases when absurdity bias leads us to make overly dismissive decisions.

7. Contagion Heuristic

Quick Definition: We can sometimes see people, ideas, and things as being either positively or negatively contagious despite lack of logic.

Sometimes, people will try to avoid contact with something or someone that has been the victim of bad luck. For example, a person may feel uncomfortable touching a cancer patient despite the fact they are not at all contagious.

On the positive end, we may believe lucky people will remain lucky and may even spread good luck if we spend time with them. Sometimes, this could be called the halo effect and horns effect.

8. Effort Heuristic

Quick Definition: Assuming the quality of something correlates with the amount of effort put into it.

We will often think something is more valuable or higher quality if it took a great deal of effort to create it. This assumption may be correct, but it doesn’t always turn out to be true.

For example, a person may spend 20 hours a day, 365 days a year, working on a startup business and it may still fail due to flaws in the business model. Another person may build a business in a week and see instant success.

Here, there is no positive correlation between effort and quality.

Nevertheless, the effort heuristic is utilized by advertisers all the time. Advertisements might talk about the amount of hours spent testing products, the research and development money put into it, and so on, in order to show that a lot of effort was put into it. The insinuation here is that the effort has led to a higher-quality product, when this is not necessarily always true.

9. Familiarity Heuristic

Quick Definition: We can often take mental shortcuts where we decide things that are most familiar to us are better than things that are less familiar.

Humans tend to see safety in the familiar and risk in the unfamiliar. In reality, familiar things may be just as risky, if not more, than unfamiliar things. Nevertheless, we know how to navigate familiar situations and therefore find them less risky.

A good example of this is travel. We may look to a country overseas and see it as potentially dangerous or scary. But, looking at data, our hometown or home city may be far more dangerous!

Similarly, we’re much more likely to die in a car crash than a plane crash. Nevertheless, fear may overcome you getting on a plane despite the fact that you didn’t put a moment’s thought into the drive to the airport.

10. Fluency Heuristic

Quick Definition: If an idea is communicated more fluently or skillfully then it is given more credence than an idea that is clumsily communicated, regardless of the merit of the idea.

The fluency with which an idea is communicated can directly impact how we perceive the idea. This mental shortcut allows us to bypass direct assessment of the merits of a case. Instead, we rely more on the charisma of the communicator.

For example, leaders with charismatic authority can often command a high vote during elections because of their ability to connect with voters moreso than their actual policy positions.

11. Gaze Heuristic

Quick Definition: Animals and humans have developed the ability to fixate on an estimated position rather than conducting complex calculations. Generally, this is in relation to motion.

The most common example of the gaze heuristic is the process humans go through to estimate where a ball will land. We don’t do all the calculations to understand trajectory and angle. Instead, we’ve developed an uncanny ability to identify where the ball will land through mental shortcuts based on previous experience.

Similarly, predatory animals can predict where their prey will flee to in order to intercept it, bats can use it during echolocation to estimate the location of obstacles, and hockey goalkeepers can use it to estimate the eventual position of a puck flying towards the goals.

12. Recognition Heuristic

Quick Definition: We assume that things we recognize have more value than things we do not recognize.

Recognition is an important facet of product marketing. Brand recognition alone can help a brand to thrive among a field of other products on a shelf.

The recognition heuristic states that we take mental shortcuts when looking at a range of options by assuming that the most recognizable option holds greater value. Thus, we assume a well-known household brand is higher-quality than a lesser-known brand.

Similarly, a study in psychology found that people assume cities whose names they recognize have larger populations than those that they don’t recognize. This assumption is based on the mental shortcut that larger cities are more likely to have recognizable names than smaller cities. This mental shortcut is often accurate, showing how heuristics can be beneficial (we call this the “less is more effect”).

13. Scarcity Heuristic

Quick Definition: When something is scarce , we see it as more valuable.

False scarcity is a widely-utilized method in marketing psychology because it encourages consumers to see a product as having greater value than it really does.

When a product is framed as being scarce, it is seen as having value because only a certain number of people can have it. As a result, people want it more. Sometimes, we call this the framing effect .

One way marketers use false scarcity is that they create limited-time discounts. In this case, the low price is a point of scarcity. Another way they can create false scarcity is to have open and closed cart periods so the product is only available for a short period of time.

This is a heuristic because people are encouraged to bypass making cold contemplative decisions about the product and, instead, make rushed decisions based on fear of missing out.

14. Similarity Heuristic

Quick Definition: Similarity between past and present situations impacts decision-making, allowing people to bypass making objective comparisons of two alternatives.

We tend to rely on past experiences to shape future experiences. If we liked something previously, then we may seek out similar situations in the future. If we didn’t like it in the past ,then we may avoid those situations in the future.

This logic allows people to bypass a thorough assessment of something and, instead, make fast decisions based on past experience.

Marketers can take advantage of this tendency. For example, a new fast food restaurant may use colors and a menu similar to McDonald;s in order to lull consumers into seeing the restaurant as similar to their previous positive experiences at McDonald’s, and therefore more likely to give it a go.

Similarly, Netflix may show you shows and movies similar to previous ones you watched to the end, because Netflix knows that you are going to be partisan toward a similar experience to the ones you previously enjoyed.

15. Simulation Heuristic

Quick Definition: We tend to overestimate the likelihood of an event based upon how easy it is to visualize it.

If our minds are able to visualize something happening, then we overstimate its probability.

Generally, the simulation heuristic occurs in relation to regret or near misses. A great example of this is buying a lottery ticket. If you found out that someone bought a winning lottery ticket one hour after you bought your ticket, then you’d easily be able to visualize the potentiality that you had gotten stuck in traffic that day and turned up to buy the ticket an hour later.

In this example, the probability of you ever turning up to buy the lottery ticket at the right time and place remains extremely low. However, because you can so easily visualize that eventuality, it feels as if you were truly very close to winning the lottery.

16. Social Proof Heuristic

Quick Definition: We use social proof as a mental shortcut to verify the quality or veracity of something instead of investigating it ourselves.

The social proof heuristic occurs both in social norms and product marketing.

In social norms, people tend to accept something as normal, correct, or appropriate because the rest of society does.

We could imagine, for example, 200 years ago many people thought the idea of the women’s right to vote as an idea that is strange or worthy of serious critique before being implemented. There weren’t many people supportive of the idea, so it was unquestioned. Today, because women’s right to vote is a social norm, it seems absurd that anyone would take it away.

In both of the above situations, people relied on broader society’s views (i.e. social proof) as an anchoring point for their own thinking on the topic.

Similarly, in marketing, marketers often go to great lengths to get quotes from “average joes” who have used a product in order to provide social proof in their advertisements.

17. Authority Heuristic

Quick Definition: We tend to defer to authorities as a shortcut rather than doing the thinking and research ourselves.

Society is structured in such a way that we defer to authorities and experts constantly. For example, we will defer to doctors on medical issues, engineers when building bridges, and lawyers on legal issues.

It’s just impossible to go about life trying to be an expert and authority on every topic. Instead, we will need to team up with authorities to make intelligent decisions. So, this heuristic is necessary.

However, mistakes can often be made when we see a person as an authority in one topic and, therefore, assume they’re an authority in entirely unrelated topics.

18. Hot-Hand Fallacy

Quick Definition: We overestimate our chances of success after a string of recent successes.

The hot-hand fallacy assumes that successful people will continue to experience success in the future.

The phrase “hot-hand” refers to gambling where a person rolling a dice has a “hot-hand” if they keep rolling the right numbers.

But we can apply this concept to a range of other situations. For example, we can apply it to investment funds, where investors will invest in a fund if it recently saw a lot of success.

However, past success does not guarantee future results. The more important thing would be to look at their investment philosophy rather than take the mental shortcut of “if they have recently been successful, then they will be in the future, too.”

19. Occam’s Razor

Quick Definition: The assumption that the most straightforward explanation is the most accurate.

Occam’s razor refers to the preferencing of more straightforward explanations as opposed to more complex ones. One logical justification for this is that the straightforward explanation has the least possible variables where mistakes in logic can occur.

However, critics of this approach highlight that, by definition, Occam’s razor fails to contemplate all possible variables and therefore causes oversimplification of explanations. Nevertheless, invoking Occam’s razor allows people to step back from a situation and contemplate whether they have over-complicated a simple situation.

>Check out these 15 occam’s razor examples

20. Naive Diversification

Quick Definition: Longer-term planning tends to involve more diversification than shorter-term planning.

Consider a situation where you are asked to purchase 5 weeks’ worth of groceries at once. In this situation, you’re more likely to buy a diverse range of fruit and vegetables for the forthcoming five weeks.

By contrast, if you were to go shopping once a week for five weeks, you’re less likely to diversify. Rather, you would buy a narrow range of products that you want in the short term.

In this example, people tend to diversify when faced with longer-term plans than shorter-term plans.

Naive diversification teaches us a lesson in business and investment. It teaches us that sometimes we are too soon to diversify when making plans because of our inability to make longer-term decisions in the shorter-term. As a result, we try to hedge by diversifying.

21. Peak–End Rule

Quick Definition: People tend to remember and pass judgment on an event based upon its most intense moment of finality rather than the average.

The peak-end rule refers to situations where the peak and end of a situation are the most important in our memories. When describing situations in the past tense, our minds shortcut to the peak and the end and fail to contemplate the other parts of the memory.

For example, a book or movie may be boring for 75% of the film, but the last 25% are excellent. You then go away and tell people how excellent it was, forgetting that there were long boring periods.

This is because our minds are most stimulated at the highly emotive parts of a situation, searing them in our memories.

This rule can be applied in vacation packages, movies, and other experince-based services where the experience is curated so the peak (and end) are highly stimulating to create a ‘wow experience’ that shapes people’s memories.

22. Mere Exposure Effect

Quick Definition: The mere exposure effect occurs when people develop a preference for a stimulus (such as a brand) simply because it is familiar. It is sometimes referred to as the familiarity principle.

The more frequently a person sees, experiences, or is otherwise exposed to something, the more likely it is that they will begin to like and favor it.

This is a cognitive heuristic because it involves a mental shortcut where something that is familiar is assumed to be safer and more trustworthy than unfamiliar things, regardless of the facts of the case.

This is used extensively in advertising, for example, where repeated exposure to advertisements from a particular brand, such as a restaurant, might make people more inclined to go to that restaurant next time they are hungry.

>See our full article on the Mere Exposure Effect

Heuristics are rules of thumb that help us make decisions quickly. They are useful in many situations, and in fact have helped us evolutionarily by filtering out bad information and making decisions quickly.

However, they can can also lead to biases and errors in our thinking. In the worst-case scenarios they can lead to stereotyping and significant social harm. The most common types of heuristics are availability heuristics, representativeness heuristics, and anchoring and adjustment.

Knowing about these biases in our thinking can help marketers to sell products and help reflective people to make better decisions by knowing when and when not to use heuristics.

See Also: Fundamental Attribution Error Examples

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2 thoughts on “22 Heuristics Examples (The Types of Heuristics)”

Really interesting reading about the types of Heuristics and thank you for the concise explanations.

The information was very concise is used it for a presentation I had. Thank you

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Reviewed by Psychology Today Staff

A heuristic is a mental shortcut that allows an individual to make a decision, pass judgment, or solve a problem quickly and with minimal mental effort. While heuristics can reduce the burden of decision-making and free up limited cognitive resources, they can also be costly when they lead individuals to miss critical information or act on unjust biases.

• Understanding Heuristics
• Different Heuristics
• Problems with Heuristics

As humans move throughout the world, they must process large amounts of information and make many choices with limited amounts of time. When information is missing, or an immediate decision is necessary, heuristics act as “rules of thumb” that guide behavior down the most efficient pathway.

Heuristics are not unique to humans; animals use heuristics that, though less complex, also serve to simplify decision-making and reduce cognitive load.

Generally, yes. Navigating day-to-day life requires everyone to make countless small decisions within a limited timeframe. Heuristics can help individuals save time and mental energy, freeing up cognitive resources for more complex planning and problem-solving endeavors.

The human brain and all its processes—including heuristics— developed over millions of years of evolution . Since mental shortcuts save both cognitive energy and time, they likely provided an advantage to those who relied on them.

Heuristics that were helpful to early humans may not be universally beneficial today . The familiarity heuristic, for example—in which the familiar is preferred over the unknown—could steer early humans toward foods or people that were safe, but may trigger anxiety or unfair biases in modern times.

The study of heuristics was developed by renowned psychologists Daniel Kahneman and Amos Tversky. Starting in the 1970s, Kahneman and Tversky identified several different kinds of heuristics, most notably the availability heuristic and the anchoring heuristic.

Since then, researchers have continued their work and identified many different kinds of heuristics, including:

Familiarity heuristic

Fundamental attribution error

Representativeness heuristic

Satisficing

The anchoring heuristic, or anchoring bias , occurs when someone relies more heavily on the first piece of information learned when making a choice, even if it's not the most relevant. In such cases, anchoring is likely to steer individuals wrong .

The availability heuristic describes the mental shortcut in which someone estimates whether something is likely to occur based on how readily examples come to mind . People tend to overestimate the probability of plane crashes, homicides, and shark attacks, for instance, because examples of such events are easily remembered.

People who make use of the representativeness heuristic categorize objects (or other people) based on how similar they are to known entities —assuming someone described as "quiet" is more likely to be a librarian than a politician, for instance. 

Satisficing is a decision-making strategy in which the first option that satisfies certain criteria is selected , even if other, better options may exist.

Heuristics, while useful, are imperfect; if relied on too heavily, they can result in incorrect judgments or cognitive biases. Some are more likely to steer people wrong than others.

Assuming, for example, that child abductions are common because they’re frequently reported on the news—an example of the availability heuristic—may trigger unnecessary fear or overprotective parenting practices. Understanding commonly unhelpful heuristics, and identifying situations where they could affect behavior, may help individuals avoid such mental pitfalls.

Sometimes called the attribution effect or correspondence bias, the term describes a tendency to attribute others’ behavior primarily to internal factors—like personality or character— while attributing one’s own behavior more to external or situational factors .

If one person steps on the foot of another in a crowded elevator, the victim may attribute it to carelessness. If, on the other hand, they themselves step on another’s foot, they may be more likely to attribute the mistake to being jostled by someone else .

Listen to your gut, but don’t rely on it . Think through major problems methodically—by making a list of pros and cons, for instance, or consulting with people you trust. Make extra time to think through tasks where snap decisions could cause significant problems, such as catching an important flight.

Your plans should not be set based on the wisdom shared by successful people. They should be based on your context, your strengths, and your creative inspiration from that wisdom.

Discover how the anchoring effect, a subtle cognitive bias, shapes our decisions across life's domains.

Artificial intelligence already plays a role in deciding who’s getting hired. The way to improve AI is very similar to how we fight human biases.

Think you are avoiding the motherhood penalty by not having children? Think again. Simply being a woman of childbearing age can trigger discrimination.

Psychological experiments on human judgment under uncertainty showed that people often stray from presumptions about rational economic agents.

Psychology, like other disciplines, uses the scientific method to acquire knowledge and uncover truths—but we still ask experts for information and rely on intuition. Here's why.

We all experience these 3 cognitive blind spots at work, frequently unaware of their costs in terms of productivity and misunderstanding. Try these strategies to work around them.

How do we make sense of new experiences? Ultimately, it's about how we categorize them—which we often do by "lumping" or "splitting" them.

Have you ever fallen for fake news? This toolkit can help you easily evaluate whether a claim is real or phony.

An insidious form of prejudice occurs when a more powerful group ignores groups with less power and keeps them out of the minds of society.

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Types of Heuristics in Psychology

Categories Cognition

When you are trying to solve a problem or make a decision, you don’t always have time to examine every possible answer or possibility. Sometimes, you have to rely on the information you already have to make the best guess or estimate in a limited amount of time.

This is an example of using heuristics, or mental ‘rules of thumb,’ that help you make choices quickly and easily. 

There are many different ways to solve problems, but some take more time than others. Heuristics can be thought of as mental shortcuts that often help us make educated guesses.

Definition : A heuristic in psychology refers to a mental shortcut or rule of thumb that helps individuals make decisions or solve problems more efficiently. It is a cognitive strategy that allows us to simplify complex information and make judgments quickly. 

Table of Contents

3 Key Types of Heuristics

There are several different types of heuristics that we commonly use in our everyday lives. 

Availability Heuristic

This heuristic involves making judgments based on the ease with which examples or instances come to mind. For example, if we can easily recall instances of successful outcomes from a particular option, we are more likely to choose it. 

Representativeness Heuristic

This heuristic involves making judgments based on how well an object or event matches a particular prototype or stereotype. For example, if someone fits our mental image of a successful entrepreneur, we may assume they are more likely to be successful. 

Anchoring and Adjustment Heuristic

The anchoring and adjustment heuristic heuristic involves starting with an initial anchor or reference point and then adjusting our judgment based on additional information. For example, when negotiating a price, we may start with a high anchor and then adjust downward based on the seller’s counteroffer. 

Why We Use Different Types of Heuristics

We don’t always have the time–or the resources–to consider every possible option for every decision we make. People make hundreds of decisions each day. If we had to meticulously use trial and error or other methods to make each choice, we’d never get anything done.

Heuristics are often used when we don’t have the time or resources to gather all the necessary information for a decision.

Of course, heuristics aren’t perfect. They can also be biased and may lead to inaccurate decisions. Despite their limitations, heuristics are valuable because they allow us to make decisions quickly and with minimal effort. 

There are different types of heuristics that individuals use in various situations. Some common types include availability heuristics, representativeness heuristics, and anchoring and adjustment heuristics. 

Each type of heuristic has its own set of characteristics and biases. Understanding heuristics is important because they can contribute to cognitive biases, which are systematic errors in thinking. By recognizing these biases, we can become more aware of our decision-making processes and make more informed choices. 

How Were These Types of Heuristics Discovered? 

While we may like to believe that our choices are rooted in rationality and logic, psychologists have shown that there are certain patterns that tend to dictate how we solve the problems we face. During the 1970s, [sychologists Amos Tversky and Daniel Kahneman conducted groundbreaking studies on how people make judgments in the face of uncertainty.

Their work challenged the traditional view that humans always make rational choices based on complete information. 

Tversky and Kahneman suggested that heuristics are mental shortcuts that individuals use to simplify complex problems and make judgments quickly. They identified several common heuristics, such as the availability heuristic, which involves making judgments based on how easily examples come to mind, and the representativeness heuristic, which involves making judgments based on how closely something resembles a typical example. 

This research provided valuable insights into the limitations and biases associated with heuristics. They showed that heuristics can lead to systematic errors in thinking known as cognitive biases . 

How We Use Different Types of Heuristics to Make Decisions?

Heuristics are an inextricable part of our daily lives, even though we are rarely aware of them. In decisions both large and small, we use heuristics to help narrow down our choices and determine which option is right for us, often based on very limited information. 

For example, the availability heuristic helps us make decisions based on the ease with which examples come to mind. If we can easily recall instances of successful outcomes from a particular option, we are more likely to choose it. 

Heuristics also aid in problem-solving by providing shortcuts to finding solutions. Instead of exhaustively analyzing every possible solution, heuristics allow us to quickly identify potential options based on past experiences or general rules of thumb. This can save time and mental effort, especially when we need to act quickly.

Advertisers often employ heuristics to create persuasive messages that appeal to consumers’ cognitive biases. By framing information in a certain way or using social proof, they can influence our decision-making and encourage specific actions. 

Examples of Types of Heuristics 

There are examples of heuristics all around us. Examining some of these examples can help give greater insight into how they shape our thinking and influence our choices. 

• For example, imagine that you are going to be flying to another country for vacation. In the weeks before your flight, you find yourself recalling numerous news stories of plane crashes. Because these examples spring to mind so readily, you may overestimate the likelihood that a plane crash will occur. This is an example of the availability heuristic.
• Or imagine you deciding who to vote for in an upcoming election. You might look at the candidates and pick one based on your expectations about good leadership traits. Basing your decision on how well the candidate fits your expectations rather than on their voting record or policy platform is an example of the representativeness heuristic.
• Or imagine that you are thinking about buying a new house. You look at the list price, and then use that number as an ‘anchor’ to base your offer on. It may not necessarily indicate what the house is worth or what other similar houses are going for, but you’re still likely to use that initial number as a reference point for all future negotiations. 

These examples highlight how heuristics can simplify decision-making but also demonstrate their potential limitations. By recognizing these heuristics in action, we can become more aware of their influence and make more informed choices. 

Heuristics vs. Other Decision-Making Strategies 

Heuristics are just one of the many strategies people utilize to make decisions. We may be more likely to rely on heuristics when:

• When need to make decisions quickly
• When we don’t have the cognitive resources to use other strategies
• When we have relevant past experience

It is important to note that heuristics and other decision-making strategies are not mutually exclusive. In fact, individuals often use a combination of heuristics and other strategies depending on the context and the specific decision at hand. 

Understanding the differences between heuristics and other decision-making strategies can help individuals become more aware of their decision-making processes and make more informed choices. By recognizing the strengths and limitations of each approach, individuals can develop a more balanced and effective decision-making toolkit.

Using Different Types of Heuristics to Make Better Decisions

Using various types of heuristics can be a valuable tool for making better decisions. By understanding how heuristics work and being aware of their potential biases, individuals can harness the power of heuristics to improve their decision-making processes. 

• Recognize when they are appropriate . Heuristics are particularly useful when time is limited or when there is a lack of information. Mental shortcuts can help individuals make quick and efficient decisions in these cases. 
• Combine them with other decision-making strategies . While heuristics provide shortcuts, they are not foolproof and can lead to cognitive biases. By incorporating other strategies such as weighing pros and cons or conducting research, individuals can mitigate the potential biases associated with heuristics. 
• Be aware of the specific heuristics being used and their potential limitations . Different types of heuristics, such as availability heuristics or representativeness heuristics, have their own biases and may not always lead to accurate judgments. 

By understanding these biases, individuals can make more informed decisions and avoid common pitfalls. In conclusion, heuristics can be a powerful tool for making better decisions. Recognizing when to use them, combining them with other strategies, and being aware of their limitations can help you make better decisions.

Bobadilla-Suarez, S., & Love, B. C. (2018). Fast or frugal, but not both: Decision heuristics under time pressure . Journal of Experimental Psychology: Learning, Memory, and Cognition , 44(1), 24–33. https://doi.org/10.1037/xlm0000419

Lindström, B., Jangard, S., Selbing, I., & Olsson, A. (2018). The role of a “common is moral” heuristic in the stability and change of moral norms . Journal of Experimental Psychology: General , 147(2), 228–242. https://doi.org/10.1037/xge0000365

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Some Helpful Problem-Solving Heuristics

A  heuristic  is a thinking strategy, something that can be used to tease out further information about a problem and thus help you figure out what to do when you don’t know what to do. Here are 25 heuristics that can be useful in solving problems. They help you monitor your thought processes, to step back and watch yourself at work, and thus keep your cool in a challenging situation.

• Ask somebody else  how to do the problem. This strategy is probably the most used world-wide, though it is not one we encourage our students to use, at least not initially.
• Guess and try  (guess, check, and revise). Your first guess might be right! But incorrect guesses can often suggest a direction toward a solution. (N.B. A spreadsheet is a powerful aid in guessing and trying. Set up the relationships and plug in a number to see if you get what you want. If you don’t, it is easy to try another number. And another.)
• Restate the problem  using words that make sense to you. One way to do this is to explain the problem to someone else. Often this is all it takes for the light to dawn.
• Organize information  into a table or chart. Having it laid out clearly in front of you frees up your mind for thinking. And perhaps you can use the organized data to generate more information.
• Draw a picture  of the problem. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or some other kind of representation.
• Make a model  of the problem. The model might be a physical or mental model, perhaps using a computer. You might vary the problem information to see whether and how the model may be affected.
• Look for patterns , any kind of patterns: number patterns, verbal patterns, spatial/visual patterns, patterns in time, patterns in sound. (Some people define mathematics as the science of patterns.)
• Act out the problem , if it is stated in a narrative form. Acting it out can have the same effect as drawing a picture. What’s more, acting out the problem might disclose incorrect assumptions you are making.
• Invent notation . Name things in the problem (known or unknown) using words or symbols, including relationships between problem components.
• Write equations . An equation is simply the same thing named two different ways.
• Check all possibilities  in a systematic way. A table or chart may help you to be systematic.
• Work backwards  from the end condition to the beginning condition. Working backwards is particularly helpful when letting a variable (letter) represent an unknown.
• Identify subgoals  in the problem. Break up the problem into a sequence of smaller problems (“If I knew this, then I could get that”).
• Simplify the problem . Use easier or smaller numbers, or look at extreme cases (e.g., use the minimum or maximum value of one of the varying quantities).
• Restate the problem again . After working on the problem for a time, back off a bit and put it into your own words in still a different way, since now you know more about it.
• Change your point of view . Use your imagination to change the way you are looking at the problem. Turn it upside down, or pull it inside out.
• Check for hidden assumptions  you may be making (you might be making the problem harder than it really is). These assumptions are often found by changing the given numbers or conditions and looking to see what happens.
• Identify needed and given information clearly . You may not need to find everything you think you need to find, for instance.
• Make up your own technique . It is your mind, after all; use mental actions that make sense to you. The key is to do something that engages you with the problem.
• Try combinations of the above heuristics .

These heuristics can be readily pointed out to students as they engage problems in the classroom. However, real-world problems are often confronted many times over or on increasingly complex levels. For those kinds of problems, George Polya, the father of modern problem-solving heuristics, identified a fifth class (E) of looking-back heuristics. We include these here for completeness, but also with the teaching caveat that solutions often improve and insights grow deeper after the initial pressure to produce a solution has been resolved. Subsequent considerations of a problem situation are invariably deeper than the first attempt.

• Check your solution . Substitute your answer or results back into the problem. Are all of the conditions satisfied?
• Find another solution . There may be more than one answer. Make sure you have them all.
• Solve the problem a different way . Your first solution will seldom be the best solution. Now that the pressure is off, you may readily find other ways to solve the problem.
• Solve a related problem . Steve Brown and Marion Walter in their book,  The Art of Problem Posing , suggest the “What if not?” technique. What if the train goes at a different speed? What if there are 8 children, instead of 9? What if . . .? Fascinating discoveries can be made in this way, leading to:
• Generalize the solution . Can you glean from your solution how it can be made to fit a whole class of related situations? Can you prove your result?

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Table of Contents

What is heuristics, history of heuristics, types of heuristics, advantages of heuristics, heuristics & cognitive bias, making quick decisions with heuristics, heuristics at a glance.

Heuristics are mental shortcut techniques used to solve problems and make decisions efficiently. These techniques are used to reduce the decision making time and allow the individual to function without interrupting their next course of action.

Heuristics are a time-saving approach to solving problems and making decisions efficiently. Heuristics processes are usually used to find quick answers and solutions to problems. However, decisions based on this mindset are not always accurate. They serve as quick mental references that are used for everyday problems and experiences.

Humans and animals resort to this mindset because processing every information that comes into the brain takes time and effort. With the help of these shortcut techniques, the brain can make faster and efficient decisions despite the consequences. This is known as the accuracy effort trade-off theory. This theory works because not every decision requires the same amount of time and energy.

Hence, people use it as a means to save time. Another reason why people resort to heuristics is that the brain simply doesn’t have the capacity to process everything and so they must resort to these mental shortcuts to make quick decisions. A 2014 study 1 Mousavi, S., & Gigerenzer, G. (2014). Risk, uncertainty, and heuristics.  Journal of Business Research ,  67 (8), 1671-1678.  https://doi.org/10.1016/j.jbusres.2014.02.013 demonstrated that in case of uncertainty and a lack of information, heuristics allows a “less is more effect” wherein less information leads to more accuracy. It is worth mentioning that the applicability and usefulness of heuristics depend on the situation.

A 2011 study 2 Gigerenzer G, Gaissmaier W. Heuristic decision making. Annu Rev Psychol. 2011;62:451-82. doi: 10.1146/annurev-psych-120709-145346. PMID: 21126183. pointed out that there may be two reasons for relying on heuristics. They are:

• Individuals and organizations often rely on simple heuristics in an adaptive way
• Ignoring part of the information can lead to more accurate judgments than weighing and adding all information

Although heuristics are useful, sometimes they can be inaccurate. In case an individual relies on it too heavily, it may result in incorrect judgments or cognitive biases. Understanding commonly unfavorable heuristics and identifying situations that may affect behavior can help individuals avoid mental pitfalls. It is important to assess major problems by making a list of pros and cons. In order to avoid inaccurate decisions, you can consult trusted individuals, take time to think through things where quick decisions may cause significant problems such as catching an important flight. Hence, it is important to be mindful of the information that is being processed in the brain to make accurate decisions.

Nobel prize-winning psychologist, Herbert Simon suggested that although people attempt to make rational decisions, humans are subject to cognitive limitations. A 2013 study 3 Rachlin, H. (2003). Rational thought and rational behavior: A review of bounded rationality: The adaptive toolbox. Journal of the Experimental Analysis of Behavior, 79(3), 409-412. https://doi.org/10.1901/jeab.2003.79-409 pointed out that rational decisions involve weighing different factors such as potential costs against potential benefits. People are often limited by time to make choices as well as the amount of information we have at our disposal. Other factors that influence our thinking are overall intelligence and accuracy of perceptions.

Psychologists Amos Tversky and Daniel Kahneman proposed that cognitive biases influence how people think and the judgements people make about events. Due to these limitations, we are often forced to rely on our instinctive shortcuts i.e heuristics to make sense of the world. Simon’s research indicated that humans have a limited ability to make rational decisions. On the other hand, Tversky and Kahneman’s research 4 Kahneman, D., & Tversky, A. (1977). Prospect theory. An analysis of decision making under risk. https://doi.org/10.21236/ada045771 represented how people have specific ways to simplify the decision-making process.

Read More About Decision-Making Here

Some of the common heuristics may include the following:

1. Availability Heuristics

This involves making decisions based on the information that is readily available in one’s mind. When an individual makes a decision, they immediately refer to a number of relevant examples. Since the relevant information is readily available in their memory, they are more likely to conclude that these outcomes are common. For example, dramatic, violent deaths are usually more highly publicized and hence have higher availability.

Another instance where availability heuristics may work is if an individual is thinking about taking a trip and thinks of a number of recent airline accidents. This may lead them to think that air travel is dangerous. This may also enable them to resort to traveling by car instead. Since airline disasters came to their mind easily, the availability heuristics lead them to think that plane crashes are more common even though it may not be entirely true.

2. Representative Heuristics

This involves making a decision based on the comparison of the present situation and the most relevant mental prototype. In case an individual is trying to decide if someone is trustworthy they may compare the incident with other mental examples. For instance, an older woman sitting beside you at a train station may remind you of your grandmother. You may immediately assume that she may be kind, gentle, and trustworthy. People tend to believe in the existing mental information since the traits match up to the individual’s mental prototype.

3. Affect Heuristics

This involves making choices that are influenced by emotions that an individual is experiencing at that moment. Research 5 Finucane, M. L., Alhakami, A., Slovic, P., & Johnson, S. M. (2000). The affect heuristic in judgments of risks and benefits. Journal of Behavioral Decision Making, 13(1), 1-17. https://doi.org/10.1002/(sici)1099-0771(200001/03)13:1<1::aid-bdm333>3.0.co;2-s has demonstrated that people are more likely to view decisions as having benefits and lower risks when their mood is positive. However, negative emotions lead people to focus on the potential downfall of a decision rather than the possible benefits.

4. Satisficing Heuristics

This is a decision making strategy wherein the first option that fulfills the criteria is selected even if there are better alternatives available. Hebert Simon formulated the concept of satisficing. This theory 6 Simon, H. A. (1955). A behavioral model of rational choice. The Quarterly Journal of Economics, 69(1), 99. https://doi.org/10.2307/1884852 is used to choose one alternative from a set of alternatives in situations of uncertainty. In this case, uncertainty refers to the total set of alternatives and their consequences that cannot be known or foreseen. For instance, professional real estate entrepreneurs rely on this theory to decide where they should invest to develop new commercial areas. Although there may be better alternatives available, they resort to the first option that fulfills their criteria.

Some of the most common advantages of using this cognitive approach are:

• Facilitates timely decisions
• Makes decision making simpler
• Less information, more accuracy
• Quick answers to problems
• Reduces complex information into simple and manageable set of choices
• Frees up cognitive resources for more complex planning

Although heuristics can advance our problems and decision-making process, it can even cause errors. It can often lead to inaccurate judgments based on how common things can occur and how certain events influence our decisions. It is important to realize that even though something worked in the past, it doesn’t necessarily mean that it will work again. Relying on existing heuristics can make it difficult to see alternatives or brainstorm new ideas. A 2014 study pointed out that heuristics can also contribute to other things such as stereotypes and prejudice. Due to this people often overlook more relevant information and create stereotypical categorization that is not entirely true.

Read More About Cognitive Bias Here

Heuristics allow us to make quick decisions and make our life easier. It is often accurate. However, it is important to be aware of what is influencing our decisions in order to avoid potential cognitive biases. This will allow us to make more accurate decisions.

• Heuristics are mental shortcut techniques used to solve problems and make decisions efficiently.
• Heuristics processes are usually used to find quick answers and solutions to problems.
• They serve as quick mental references that are used for everyday problems and experiences.
• With the help of these shortcut techniques, the brain can make faster and efficient decisions despite the consequences.
• Sometimes it may result in incorrect judgments or cognitive biases.
• It is important to be aware of what is influencing our decisions in order to avoid potential cognitive biases.

References:

• 1 Mousavi, S., & Gigerenzer, G. (2014). Risk, uncertainty, and heuristics.  Journal of Business Research ,  67 (8), 1671-1678.  https://doi.org/10.1016/j.jbusres.2014.02.013
• 2 Gigerenzer G, Gaissmaier W. Heuristic decision making. Annu Rev Psychol. 2011;62:451-82. doi: 10.1146/annurev-psych-120709-145346. PMID: 21126183.
• 3 Rachlin, H. (2003). Rational thought and rational behavior: A review of bounded rationality: The adaptive toolbox. Journal of the Experimental Analysis of Behavior, 79(3), 409-412. https://doi.org/10.1901/jeab.2003.79-409
• 4 Kahneman, D., & Tversky, A. (1977). Prospect theory. An analysis of decision making under risk. https://doi.org/10.21236/ada045771
• 5 Finucane, M. L., Alhakami, A., Slovic, P., & Johnson, S. M. (2000). The affect heuristic in judgments of risks and benefits. Journal of Behavioral Decision Making, 13(1), 1-17. https://doi.org/10.1002/(sici)1099-0771(200001/03)13:1<1::aid-bdm333>3.0.co;2-s
• 6 Simon, H. A. (1955). A behavioral model of rational choice. The Quarterly Journal of Economics, 69(1), 99. https://doi.org/10.2307/1884852

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Home Blog Business Using Heuristic Problem-Solving Methods for Effective Decision-Making

Using Heuristic Problem-Solving Methods for Effective Decision-Making

Problem-solving capability and effective decision making are two of the most prized capabilities of any leader. However, one cannot expect these traits to be simply present by default in an individual, as both require extensive analysis of the root cause of issues and to know what to look for when anticipating a gain. In a previous article, we brought you  5 Problem-Solving Strategies to Become a Better Problem Solver . This time we have something that can help you dig deep to resolve problems, i.e. using heuristic problem-solving methods for effective decision-making.

What are Heuristics?

Heuristics are essentially problem-solving tools that can be used for solving non-routine and challenging problems. A heuristic method is a practical approach for a short-term goal, such as solving a problem. The approach might not be perfect but can help find a quick solution to help move towards a reasonable way to resolve a problem.

Example: A computer that is to be used for an event to allow presenters to play PowerPoint presentations via a projector malfunctions due to an operating system problem. In such a case a system administrator might quickly refresh the system using a backup to make it functional for the event. Once the event concludes the system administrator can run detailed diagnostic tests to see if there are any further underlying problems that need to be resolved.

In this example, restoring the system using a backup was a short-term solution to solve the immediate problem, i.e. to make the system functional for the event that was to start in a few hours. There are a number of heuristic methods that can lead to such a decision to resolve a problem. These are explained in more detail in the sections below.

Examples of Heuristic Methods Used for Challenging and Non-Routine Problems

Heuristic methods can help ease the cognitive load by making it easy to process decisions. These include various basic methods that aren’t rooted in any theory per se but rather rely on past experiences and common sense. Using heuristics one can, therefore, resolve challenging and non-routine problems. Let’s take a look at some examples.

A Rule of Thumb

This includes using a method based on practical experience. A rule of thumb can be applied to find a short-term solution to a problem to quickly resolve an issue during a situation where one might be pressed for time.

Example: In the case of the operating system failure mentioned earlier, we assume that the PC on which PowerPoint presentations are to be run by presenters during an event is getting stuck on the start screen. Considering that the event is about to start in 2 hours, it is not practical for the system administrator to reinstall the operating system and all associated applications, hotfixes and updates, as it might take several hours. Using a rule of thumb, he might try to use various tried and tested methods, such as trying to use a system restore point to restore the PC without deleting essential files or to use a backup to restore the PC to an earlier environment.

An Educated Guess

An educated guess or guess and check can help resolve a problem by using knowledge and experience. Based on your knowledge of a subject, you can make an educated guess to resolve a problem.

Example: In the example of the malfunctioning PC, the system administrator will have to make an educated guess regarding the best possible way to resolve the problem. The educated guess, in this case, can be to restore the system to a backup instead of using system restore, both of which might take a similar amount of time; however, the former is likely to work better as a quick fix based on past experience and knowledge of the system administrator.

Trial and Error

This is another heuristic method to problem-solving where one might try various things that are expected to work until a solution is achieved.

Example: The system administrator might try various techniques to fix the PC using trial and error. He might start with checking if the system is accessible in safe mode. And if so, does removing a newly installed software or update solve the problem? If he can’t access the system at all, he might proceed with restoring it from a backup. If that too fails, he might need to quickly opt for a wipe and load installation and only install PowerPoint to ensure that at least presenters can run presentations on the PC. In this case he can perform other required software installations after the event.

An Intuitive Judgment

Intuitive judgment does not result from a rational analysis of a situation or based on reasoning. It is more of a feeling one has which may or may not lead to the desired outcome. Sometimes, intuitive judgement can help resolve problems. Perhaps the most rational way to describe an intuition is that it is some type of calculation at the subconscious level, where you can’t put your finger on the reason why you think something might be the way it is.

Example: The system administrator might have a feeling that the PC is not working because the hard drive has failed. This might be an intuitive judgment without hard evidence. He might quickly replace the hard drive to resolve the problem. Later, after he runs diagnostics on the old hard drive, he might realize that it was indeed that hard drive that was faulty and trying to fix it would have been a waste of time. In this case, he might be able to solve a problem using intuitive judgment.

Stereotyping

A stereotype is an opinion which is judgmental rather than rational. Certain types of possessions for example create a stereotype of social status. A person who wears an expensive watch might be deemed rich, although he might simply have received it as a gift from someone, instead of being rich himself.

Example: A certain company might have developed a bad reputation of developing faulty hard drives. If the systems administrator sees the name of that company on the hard drive when opening the faulty PC, he might think that the hard drive is faulty based on stereotyping and decide to replace it.

Profiling is used to systematically analyze data to understand its dynamics. Profiling as a heuristic method for problem-solving might entail analyzing data to understand and resolve a problem or to look for patterns, just like a root cause analysis .

Example: To solve the issue of the faulty PC, a system administrator might look for similar patterns which might have led to the problem. He might search online for solutions via online forums to understand what might have caused the issue. He might also look at the information associated with recently installed software and updates to see if something conflicted with the operating system. During the profiling process, he might realize that software he installed yesterday before shutting down the PC is the cause of the problem, since similar issues have been reported by other users. He might try to remove the software using Safe Mode or by removing its files by running the computer from a bootable disc drive.

Common Sense

Common sense is the use of practical judgment to understand something. The use of common sense is also a heuristic method used for problem-solving.

Example: When dealing with a faulty PC the system administrator sees smoke coming out of the PC. In this case, it is common sense that a hardware component is faulty. He shuts down the PC, removes the power cord and investigates the issue further based on common sense. This is because keeping the system linked to a power socket amidst smoke emitting from the PC can only make things worse. It is common sense to turn off everything and take the necessary precautions to investigate the issue further.

How are Heuristic Methods Used in Decision-Making?

There are a number of formal and informal models of heuristics used for decision making. Let’s take a look at a few of the formal models of heuristics used for decision making.

Formal Models of Heuristics

Fast-and-frugal tree.

A fast-and-frugal tree is a classification or decision tree. It is a graphical form that helps make decisions. For example, a fast-and-frugal tree might help doctors determine if a patient should be sent to a regular ward or for an emergency procedure. fast-and-frugal trees are methods for making decisions based on hierarchical models, where one has to make a decision based on little information.

Fluency Heuristic

In psychology, fluency heuristic implies an object that can be easily processed and deemed to have a higher value, even if it is not logical to assume this. Understanding the application of fluency heuristic can help make better decisions in a variety of fields. Fluency heuristic is more like sunk cost fallacy .

For example, a designer might design a user interface that is easier for users to process, with fewer buttons and easily labeled options. This can help them think fast, work quicker and improve productivity. Similarly, the concept might be used in marketing to sell products using effective marketing techniques. Even if two products are identical, a consumer might pick one over the other based on fluency heuristic. The consumer might deem the product to be better for his needs, even if it is the same as the other one.

Gaze Heuristic

Assume that you aim to catch a ball. Based on your judgment you would leap to catch the ball. If you were to leave yourself to instinct, you will end up at the same spot to catch the ball at a spot you would predict it to fall. This is essentially gaze heuristic. The concept of gaze heuristic is thought to be applied for simple situations and its applications are somewhat limited.

Recognition Heuristic

If there are two objects, one recognizable and the one isn’t, the person is likely to deem the former to be of greater value. A simple example of recognition heuristic is branding. People get used to brand logos, assuming them to be of high quality. This helps brands to sell multiple products using recognition heuristic. So, if you are looking to buy an air conditioner and come across two products, A and B, where A is a brand you know and B is a new company you don’t recognize, you might opt for A. Even if B is of better quality, you might simply trust A because you have been buying electronics from the brand for many years and they have been of good quality.

Satisficing

Satisficing entails looking for alternatives until an acceptable threshold can be ensured. Satisficing in decision making implies selecting an option which meets most needs or the first option which can meet a need, even if it is not the optimal solution. For example, when choosing between early retirement or continuing service for 2 or 3 more years, one might opt for early retirement assuming that it would meet the individual’s needs.

Similarity Heuristic

Similarity heuristic is judgment based on which is deemed similar, if something reminds someone of good or bad days, something similar might be considered the same. Similarity heuristics is often used by brands to remind people of something that they might have sentimental value for.

Someone might buy a limited-edition bottle of perfume that is being sold in a packaging style that was replaced 20 years ago. Assuming that sales were great in those days, the company might sell such limited-edition perfume bottles in the hope of boosting sales. Consumers might buy them simply because they remind them of the ‘good old days’, even though the product inside might not even be of the same but rather similar to what it used to be. Many consumers claim to buy these types of products claiming that it reminds them of a fond memory, such as their youth, marriage or  first job, when they used the product back in the day.

Final Words

Heuristics play a key role in decision making and affect the way we make decisions. Understanding heuristics can not only help resolve problems but also understand biases that affect effective decision making. A business decision or one that affects one’s health, life, or well-being cannot rely merely on a hunch. Understanding heuristics and applying them effectively can therefore help make the best possible decisions. Heuristic methods are not only used in different professions and personal decision making but are also used in artificial intelligence and programming.

Modern anti-virus software for instance uses heuristic methods to dig out the most elusive malware. The same rule can be essentially applied to decision making, by effectively using heuristics to resolve problems and to make decisions based on better judgment.

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What Are Heuristics?

Understanding heuristics.

• Pros and Cons
• Examples in Behavioral Economics

Heuristics and Psychology

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Heuristics: Definition, Pros & Cons, and Examples

James Chen, CMT is an expert trader, investment adviser, and global market strategist.

Heuristics are mental shortcuts that help people make quick decisions. They are rules or methods that help people use reason and past experience to solve problems efficiently. Commonly used to simplify problems and avoid cognitive overload, heuristics are part of how the human brain evolved and is wired, allowing individuals to quickly reach reasonable conclusions or solutions to complex problems. These solutions may not be optimal ones but are often sufficient given limited timeframes and calculative capacity.

These cognitive shortcuts feature prominently in behavioral economics .

Key Takeaways

• Heuristics are mental shortcuts for solving problems in a quick way that delivers a result that is sufficient enough to be useful given time constraints.
• Investors and financial professionals use a heuristic approach to speed up analysis and investment decisions.
• Heuristics can lead to poor decision-making based on a limited data set, but the speed of decisions can sometimes make up for the disadvantages.
• Behavioral economics has focused on heuristics as one limitation of human beings behaving like rational actors.
• Availability, anchoring, confirmation bias, and the hot hand fallacy are some examples of heuristics people use in their economic lives.

Investopedia / Danie Drankwalter

People employ heuristics naturally due to the evolution of the human brain. The brain can only process so much information at once and therefore must employ various shortcuts or practical rules of thumb . We would not get very far if we had to stop to think about every little detail or collect every piece of available information and integrate it into an analysis.

Heuristics therefore facilitate timely decisions that may not be the absolute best ones but are appropriate enough. Individuals are constantly using this sort of intelligent guesswork, trial and error, process of elimination, and past experience to solve problems or chart a course of action. In a world that is increasingly complex and overloaded with big data, heuristic methods make decision-making simpler and faster through shortcuts and good-enough calculations.

First identified in economics by the political scientist and organizational scholar Herbert Simon in his work on bounded rationality, heuristics have now become a cornerstone of behavioral economics.

Rather than subscribing to the idea that economic behavior was rational and based upon all available information to secure the best possible outcome for an individual ("optimizing"), Simon believed decision-making was about achieving outcomes that were "good enough" for the individual based on their limited information and balancing the interests of others. Simon called this " satisficing ," a portmanteau of the words "satisfy" and "suffice."

Advantages and Disadvantages of Using Heuristics

The main advantage to using heuristics is that they allow people to make good enough decisions without having all of the information and without having to undertake complex calculations.

Because humans cannot possibly obtain or process all the information needed to make fully rational decisions, they instead seek to use the information they do have to produce a satisfactory result, or one that is good enough. Heuristics allow people to go beyond their cognitive limits.

Heuristics are also advantageous when speed or timeliness matters—for example, deciding to enter a trade or making a snap judgment about some important decision. Heuristics are thus handy when there is no time to carefully weigh all options and their merits.

Disadvantages

There are also drawbacks to using heuristics. While they may be quick and dirty, they will likely not produce the optimal decision and can also be wrong entirely. Quick decisions without all the information can lead to errors in judgment, and miscalculations can lead to mistakes.

Moreover, heuristics leave us prone to biases that tend to lead us toward irrational economic behavior and sway our understanding of the world. Such heuristics have been identified and cataloged by the field of behavioral economics.

Quick & easy

Allows decision-making that goes beyond our cognitive capacity

Allows for snap judgments when time is limited

Often inaccurate

Can lead to systemic biases or errors in judgment

Example of Heuristics in Behavioral Economics

Representativeness.

A popular shortcut method in problem-solving identified in behavioral economics is called representativeness heuristics. Representativeness uses mental shortcuts to make decisions based on past events or traits that are representative of or similar to the current situation.

Say, for example, Fast Food ABC expanded its operations to India and its stock price soared. An analyst noted that India is a profitable venture for all fast-food chains. Therefore, when Fast Food XYZ announced its plan to explore the Indian market the following year, the analyst wasted no time in giving XYZ a "buy" recommendation.

Although their shortcut approach saved reviewing data for both companies, it may not have been the best decision. Fast Food XYZ may have food that is not appealing to Indian consumers, which research would have revealed.

Anchoring and Adjustment

Anchoring and adjustment is another prevalent heuristic approach. With anchoring and adjustment, a person begins with a specific target number or value—called the anchor—and subsequently adjusts that number until an acceptable value is reached over time. The major problem with this method is that if the value of the initial anchor is not the true value, then all subsequent adjustments will be systematically biased toward the anchor and away from the true value.

An example of anchoring and adjustment is a car salesman beginning negotiations with a very high price (that is arguably well above the  fair value ). Because the high price is an anchor, the final price will tend to be higher than if the car salesman had offered a fair or low price to start.

Availability (Recency) Heuristic

The availability (or recency) heuristic is an issue where people give too much weight to the probability of an event happening again if it recently has occurred. For instance, if a shark attack is reported in the news, those headlines make the event salient and can lead people to stay away from the water, even though shark attacks remain very rare.

Another example is the case of the " hot hand ," or the sense that following a string of successes, an individual is likely to continue being successful. Whether at the casino, in the markets, or playing basketball, the hot hand has been debunked. A string of recent good luck does not alter the overall probability of events occurring.

Confirmation Bias

Confirmation bias is a well-documented heuristic whereby people give more weight to information that fits with their existing worldviews or beliefs. At the same time, information that contradicts these beliefs is discounted or rejected.

Investors should be aware of their own tendency toward confirmation bias so that they can overcome poor decision-making, missing chances, and avoid falling prey to bubbles . Seeking out contrarian views and avoiding affirmative questions are two ways to counteract confirmation bias.

Hindsight Bias

Hindsight is always 20/20. However, the hindsight bias leads us to forget that we made incorrect predictions or estimates prior to them occurring. Rather, we become convinced that we had accurately predicted an event before it occurred, even when we did not. This can lead to overconfidence for making future predictions, or regret for not taking past opportunities.

Stereotypes

Stereotypes are a kind of heuristic that allows us to form opinions or judgments about people whom we have never met. In particular, stereotyping takes group-level characteristics about certain social groups—often ones that are racist, sexist, or otherwise discriminatory—and casts those characteristics onto all of the members in that group, regardless of their individual personalities, beliefs, skills, or behaviors.

By imposing oversimplified beliefs onto people, we can quickly judge potential interactions with them or individual outcomes of those people. However, these judgments are often plain wrong, derogatory, and perpetuate social divisions and exclusions.

Heuristics were first identified and taken seriously by scholars in the middle of the 20th century with the work of Herbert Simon, who asked why individuals and firms don't act like rational actors in the real world, even with market pressures punishing irrational decisions. Simon found that corporate managers do not usually optimize but instead rely on a set of heuristics or shortcuts to get the job done in a way that is good enough (to "satisfice").

Later, in the 1970s and '80s, psychologists Amos Tversky and Daniel Kahneman working at the Hebrew University in Jerusalem, built off of Herbert Simon's work and developed what is known as Prospect Theory . A cornerstone of behavioral economics, Prospect Theory catalogs several heuristics used subconsciously by people as they make financial evaluations.

One major finding is that people are loss-averse —that losses loom larger than gains (i.e., the pain of losing $50 is far more than the pleasure of receiving $50). Here, people adopt a heuristic to avoid realizing losses, sometimes spurring them to take excessive risks in order to do so—but often leading to even larger losses.

More recently, behavioral economists have tried to develop policy measures or "nudges" to help correct people's irrational use of heuristics in order to help them achieve more optimal outcomes—for instance, by having people enroll in a retirement savings plan by default instead of having to opt in.

What Are the Types of Heuristics?

To date, several heuristics have been identified by behavioral economics—or else developed to aid people in making otherwise complex decisions. In behavioral economics, representativeness, anchoring and adjustment, and availability (recency) are among the most widely cited. Heuristics may be categorized in many ways, such as cognitive versus emotional biases or errors in judgment versus errors in calculation.

What Is Heuristic Thinking?

Heuristic thinking uses mental shortcuts—often unconsciously—to quickly and efficiently make otherwise complex decisions or judgments. These can be in the form of a "rule of thumb" (e.g., saving 5% of your income in order to have a comfortable retirement) or cognitive processes that we are largely unaware of like the availability bias.

What Is Another Word for Heuristic?

Heuristic may also go by the following terms: rule of thumb; mental shortcut; educated guess; or satisfice.

How Does a Heuristic Differ From an Algorithm?

An algorithm is a step-by-step set of instructions that are followed to achieve some goal or outcome, often optimizing that outcome. They are formalized and can be expressed as a formula or "recipe." As such, they are reproducible in the sense that an algorithm will always provide the same output, given the same input.

A heuristic amounts to an educated guess or gut feeling. Rather than following a set of rules or instructions, a heuristic is a mental shortcut. Moreover, it often produces sub-optimal and even irrational outcomes that may differ even when given the same input.

What Are Computer Heuristics?

In computer science, a heuristic refers to a method of solving a problem that proves to be quicker or more efficient than traditional methods. This may involve using approximations rather than precise calculations or techniques that circumvent otherwise computationally intensive routines.

Heuristics are practical rules of thumb that manifest as mental shortcuts in judgment and decision-making. Without heuristics, our brains would not be able to function given the complexity of the world, the amount of data to process, and the calculative abilities required to form an optimal decision. Instead, heuristics allow us to make quick, good-enough choices.

However, these choices may also be subject to inaccuracies and systemic biases, such as those identified by behavioral economics.

Simon, Herbert. " Herbert Simon, Innovation, and Heuristics ." Mind & Society, vol. 17, 2019, pp. 97-109.

Kahneman, Daniel, and Tversky, Amos. " Prospect Theory: An Analysis of Decision Under Risk ." The Econometric Society, vol. 47, no. 2, 1979, pp. 263-292.

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Exploring Heuristic Methods For Problem Solving

Heuristic methods refer to experience-based techniques for mastering problem solving , learning , and discovery that find solutions that are good enough but may not be optimal.

Heuristics are mental shortcuts that allow people to solve problems and make judgments quickly based on intuitive judgments.

Background In Heuristic Methods In Business

Heuristic methods have long been used in business as a practical approach to making decisions and judgments efficiently.

The rationale behind using heuristics in business is that they allow managers and leaders to make timely assessments and choices without getting bogged down in a lengthy optimization process.

Some key reasons why businesses rely on heuristic methods include:

• Business problems are often complex with many variables, making strict optimization difficult. Heuristics allow for quick solutions.
• There is often uncertainty and ambiguity in business situations. Heuristics help managers make decent decisions with incomplete information.
• Businesses need to be agile and adapt quickly to changing conditions. Heuristics facilitate fast decision making.
• Finding optimal solutions can require significant time, resources, and information that businesses often don’t have. Heuristics provide good enough solutions.

Examples Of Heuristic Methods Used In Business

Here are 7 common heuristic techniques used in a business context:

• Satisficing: Managers set a minimum threshold or standard and choose the first option that meets it rather than finding the optimal solution.
• Rules of Thumb: Businesses use simple decision rules gained from experience, such as setting prices at double the production cost.
• Extrapolation: Projecting historical data into the future based on current trends, such as forecasting sales.
• Estimation: Gauging an approximate value when the precise value is unknown, like estimating project costs.
• Elimination-by-Aspects: Removing options based on a key disqualifying trait rather than systematically comparing all options.
• Recognition: Making decisions by recognizing similarities of a current situation to past experiences.
• Availability : Judging an event’s likelihood based on how readily examples come to mind. Used to assess business risks.

Challenges With Relying On Heuristics

While heuristic methods are very useful in business, overreliance on heuristics can also lead to poor decisions and cognitive biases. Some limitations to keep in mind include:

• Heuristics may lead to suboptimal solutions rather than optimal ones.
• They are prone to biases which can skew judgment.
• Overuse of heuristics can inhibit creative problem solving .
• They rely heavily on the intuition and past experience of the individual.
• Changing environments can make previously useful heuristics less relevant.

Using Heuristics Positively In Business

Despite their shortcomings, heuristics can be leveraged productively in business. Here are 10 tips:

• Clearly define the problem before applying a heuristic.
• Use heuristics as one input but don’t rely entirely on them.
• Combine heuristics with optimization models when possible.
• Leverage heuristics to direct and focus optimization techniques.
• Apply multiple heuristics and look for consensus.
• Update heuristics as the business environment evolves.
• Weigh heuristic solutions against data and metrics when available.
• Recognize the influence of biases and test assumptions.
• Consider when heuristics may lead to suboptimal results or cognitive biases.
• Ensure heuristics align with organizational values and goals.

Heuristic methods provide business leaders with practical mental shortcuts for making timely decisions with limited information.

While overreliance on heuristics can be risky, used judiciously and strategically they offer a valuable decision-making tool.

Businesses should leverage heuristics while being aware of their limitations and mitigating the risks of biases. With the right approach, they can enhance agility and performance.

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Heuristic Approaches to Problem Solving

“A heuristic technique, often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgement, guesstimate, stereotyping, profiling, or common sense.” (Source: Wikipedia )

“In computer science, a heuristic is a technique designed for solving a problem more quickly when classic methods are too slow, or for finding an approximate solution when classic methods fail to find any exact solution. This is achieved by trading optimality, completeness, accuracy, or precision for speed. In a way, it can be considered a shortcut.” (Source: Wikipedia )

The objective of a heuristic algorithm is to apply a rule of thumb approach to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. There is no guarantee that the solution found will be the most accurate or optimal solution for the given problem. We often refer the solution as “good enough” in most cases.

Heuristic Algorithms? Heuristic Algorithms can be found in:

Let’s investigate a few basic examples where a heuristic algorithm can be used:

Based on this approach, can you think of how a similar approach could be used for an algorithm to play:

• Othello (a.k.a. Reversi Game)
• A Battleship game?
• Rock/Paper/Scissors?

It is hence essential to use a heuristic approach to quickly discard some moves which would most likely lead to a defeat while focusing on moves that would seem to be a good step towards a win!

Let’s consider the above scenario when investigating all the possible moves for this white pawn. Can the computer make a quick decision as to what would most likely be the best option?

Alternatively, a machine learning algorithm could play the game and record and update statistics after playing each card to progressively learn which criteria is more likely to win the round for each card in the deck. You can investigate how machine learning can be used in a game of Top Trumps by reading this blog post. Heuristic methods can be used when developing algorithms which try to understand what the user is saying, or asking for. For instance, by looking for words associations, an algorithm can narrow down the meaning of words especially when a word can have two different meanings:

e.g. When using Google search a user types: “Raspeberry Pi Hardware” We can deduct that in this case Raspberry has nothing to do with the piece of fruit, so there is no need to give results on healthy eating, cooking recipes or grocery stores…

However if the user searches for “Raspeberry Pie ingredients” , we can deduct that the user is searching for a recipe and is less likely to be interested in programming blogs or computer hardware online shops. Short Path Algorithms used by GPS systems and self-driving cars also use a heuristic approach to decide on the best route to go from A to Z. This is for instance the case for the A* Search algorithm which takes into consideration the distance as the crow flies between two nodes to decide which paths to explore first and hence more effectively find the shortest path between two nodes.

You can compare two different algorithms used to find the shortest route from two nodes of a graph:

• Dijkstra’s Shortest Path Algorithm (Without using a heuristic approach)
• A* Search Algorithm (Using a heuristic approach)

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Welcome to the daily solving of our PROBLEM OF THE DAY with Ayush Tripathi. We will discuss the entire problem step-by-step and work towards developing an optimized solution. This will not only help you brush up on your concepts of Matrix but also build up problem-solving skills. Given a binary matrix contains 0s and 1s only, we need to find the sum of coverage of all zeros of the matrix where coverage for a particular 0 is defined as a total number of ones around a zero in left, right, up and bottom directions.

Input: matrix = [[0, 1, 0],           [0, 1, 1],          [0, 0, 0]] Output : 6 Explanation: There are a total of 6 coverage are there.

Give the problem a try before going through the video. All the best!!! Problem Link: https://practice.geeksforgeeks.org/problems/coverage-of-all-zeros-in-a-binary-matrix4024/1

ORIGINAL RESEARCH article

Solving the capacitated vehicle routing problem with time windows via graph convolutional network assisted tree search and quantum-inspired computing.

• Institute of Mathematics, Hamburg University of Technology, Hamburg, Germany

Vehicle routing problems are a class of NP-hard combinatorial optimization problems which attract a lot of attention, as they have many practical applications. In recent years there have been new developments solving vehicle routing problems with the help of machine learning, since learning how to automatically solve optimization problems has the potential to provide a big leap in optimization technology. Prior work on solving vehicle routing problems using machine learning has mainly focused on auto-regressive models, which are connected to high computational costs when combined with classical exact search methods as the model has to be evaluated in every search step. This paper proposes a new method for approximately solving the capacitated vehicle routing problem with time windows (CVRPTW) via a supervised deep learning-based approach in a non-autoregressive manner. The model uses a deep neural network to assist finding solutions by providing a probability distribution which is used to guide a tree search, resulting in a machine learning assisted heuristic. The model is built upon a new neural network architecture, called graph convolutional network, which is particularly suited for deep learning tasks. Furthermore, a new formulation for the CVRPTW in form of a quadratic unconstrained binary optimization (QUBO) problem is presented and solved via quantum-inspired computing in cooperation with Fujitsu, where a learned problem reduction based upon the proposed neural network is applied to circumvent limitations concerning the usage of quantum computing for large problem instances. Computational results show that the proposed models perform very well on small and medium sized instances compared to state-of-the-art solution methods in terms of computational costs and solution quality, and outperform commercial solvers for large instances.

1. Introduction

The capacitated vehicle routing problem with time windows (CVRPTW) is a well-known combinatorial optimization problem that arises in a variety of practical contexts, including delivery scheduling, emergency response planning, and supply chain management. In the CVRPTW, a fleet of vehicles must be routed to deliver goods to a set of customers, subject to capacity constraints and time window constraints, which specify the allowable time periods for the deliveries to be made. Finding the optimal routes for the vehicles is a challenging problem, as it involves balancing the conflicting objectives of minimizing the total distance traveled and maximizing the number of deliveries that can be made within the time window constraints.

Over the years, a wide range of solution approaches have been proposed for the CVRPTW, including exact algorithms, such as branch-and-cut and branch-and-price [ 1 ], and heuristics, such as genetic algorithms, simulated annealing, and tabu search [ 2 ]. These methods differ in their complexity and the quality of the solutions they produce. In recent years, there has been a growing interest in developing approximate algorithms for the CVRPTW, as these methods can scale to large-sized instances and produce high-quality solutions in reasonable time. Examples of such algorithms include the adaptive large neighborhood search and the variable neighborhood search [ 3 , 4 ].

Heuristics for routing problems can be divided into two categories: construction heuristics and improvement heuristics [ 1 ]. Similarly, machine learning-based methods for solving routing problems also feature these characteristics. Some approaches, such as those in [ 5 ] and [ 6 ], focus on iteratively improving an existing solution, while others, such as those developed by [ 7 ], [ 8 ], and [ 9 ] generate a solution for the CVRP by adding one node at a time. Improvement approaches typically rely on an initial solution that they can then refine over time. However, it may be challenging to find a suitable starting solution for more complex problems like the CVRPTW. In fact, finding a first feasible solution for the CVRPTW with a fixed number of vehicles is an NP-hard problem on its own [ 10 ]. Furthermore, improvement approaches often require multiple iterations to arrive at good solutions, and the number of iterations required tends to escalate with an increase in problem complexity. This non-linear relationship between problem size and iteration count implies that more complex problems require substantially greater computational resources to attain optimal solutions using iterative improvement methods. On the other hand, constructive methods can generate solutions within a set number of steps that is linearly dependent on the size of the problem. But the limited information available about the other tours while constructing new tours node by node can lead to inefficient constructions and obtaining additional information in the space of solutions that are constructed sequentially becomes expensive quickly.

Moreover, there has been a surge of interest in quantum computers and the potential they hold for solving complex optimization problems, that are beyond the capabilities of classical computers. Quantum computers have the ability to perform certain types of optimization tasks much faster and more efficiently than classical computers. As quantum computers might become more advanced and accessible in the nearer future, the development of models for optimization problems that can take advantage of their unique capabilities becomes increasingly relevant.

Our goal in this work is to merge the latest advances in deep learning techniques for routing problems with quantum computing. We do this by adapting the work of [ 11 ] for the CVRPTW and creating a constructive heuristic for the CVRPTW that utilizes a deep learning model. We then proceed by developing a novel formulation of the CVRPTW as a quadratic unconstrained binary optimization problem and derive a binary quadratic program from this formulation. We then use the deep learning model as a form of learned problem reduction [ 12 ] to reduce the problem instances to a size that can be handled by quantum-inspired computers from Fujitsu and conduct computational experiments for both approaches. We show that our constructive deep learning heuristic model outperforms commercial state-of-the-art solvers such as Gurobi [ 13 ] for larger instance sizes for the CVRPTW, while being close to competitive with the highly-optimized LKH heuristic [ 14 ] and Google's OR-Tools [ 15 ] on smaller instances in terms of solution quality, showing the power deep learning has to handle difficult constraints such as time windows. But for large instances a shortcoming of the constructive nature of the model becomes apparent, as the very limited information on which the next decision is based prevents finding the best solutions. Our quadratic unconstrained binary optimization model, which is solved through quantum-inspired computing, aims at overcoming some of these challenges due to its non-constructive nature. Our results show the potential that the combination of deep learning and quantum computing holds.

The remaining paper is organized as follows: In Section 2, a comprehensive overview of related work on constructive approaches for solving the CVRPTW and variants using deep learning tools is provided. The proposed models are described in detail in Section 3. In Section 4, the computational results obtained from the developed models are presented and discussed. A summary of the results and an outlook for future research is given in Section 5.

2. Related work

Building on the work of [ 16 ], who introduced the Pointer Network (PtrNet), which is a deep neural network that uses attention to output a permutation of the input and was trained in a supervised way to solve the Traveling Salesman Problem (TSP), many improvements for this constructive approach have been proposed. An extension to reinforcement learning for using the PtrNet to solve TSPs was proposed by [ 17 ]. Nazari et al. [ 7 ] adapted this for the CVRP, but replaced the recurrent neural network part [ 18 ] of the encoder by a linear embedding layer with shared parameters. Reinforcement learning as the training strategy was also pursued in various other models for TSP variants [ 19 – 21 ] as well as for the CVRP [ 8 , 20 , 22 , 23 ], since supervised learning approaches depend on the availability of large sets of high-quality solutions, as noted in [ 24 ]. One commonality among these constructive methods is that they are auto-regressive, which means the model must be evaluated every time a new node is added to the tour.

Unlike these auto-regressive methods, Joshi et al. [ 11 ] used supervised learning to train a graph neural network to produce a tour for the TSP in the form of an adjacency matrix. This matrix is then converted into a feasible solution for the TSP using beam search, a limited-width breadth-first search [ 25 ]. They build on the work of [ 26 ], who followed a similar approach to approximately solve the TSP by using a graph neural network [ 27 ], but their model performed poorly even on smaller instances. Instead of using graph neural networks, Joshi et al. [ 11 ] use deep graph ConvNets [ 28 ], which are graph convolutional neural networks that are able to learn from larger training sets, and were able to outperform all other learning-based approaches for the TSP. This outcome is unsurprising since supervised learning techniques tend to perform better than reinforcement learning techniques when enough training data is available. Our approach builds on the work of [ 11 ].

However, incorporating time window constraints is a difficult task, and there have been only a few recent proposals for constructive approaches to the CVRPTW that utilize deep learning. The first constructive method using deep learning for the CVRPTW was proposed by [ 9 ]. They use the attention model from [ 8 ] for the CVRP, which constructs one route at a time by treating all not yet visited nodes as actions and learning a policy model to choose the next node via reinforcement learning. Falkner and Schmidt-Thieme [ 9 ] extend this approach for the CVRPTW by constructing multiple routes simultaneously and using the information of all partially constructed routes to choose the next node. Although the computational results of this model are promising, it is computationally intensive to use. Furthermore, their main objective is not to minimize the total distance traveled, but a combination of total distance traveled, waiting times for the vehicles and number of vehicles used as well as also considering soft time windows, where violating time window constraints is penalized rather than forbidden, but through appropriate weighting the objective function could be adjusted to focus on one goal. Falkner et al. [ 29 ] extend the work of [ 9 ] by replacing the self-attention layers of their model by graph neural networks to encode the problem and propose a Large Neighborhood Search using a learned construction heuristic via reinforcement learning to re-construct partially destructed solutions in an auto-regressive manner. A hierarchical reinforcement learning model based on pointer networks was proposed by [ 30 ]. This model involves learning to obtain feasible solutions at a lower level and using these solutions as input for a second decoder to minimize the total distance. However, this approach is only effective for relatively small numbers of customers. Two improvement-based methods for the CVRPTW have been proposed recently. The first one by [ 31 ] uses an enhanced version of the graph attention network [ 32 ] to learn a heuristic for Very Large-scale Neighborhood Search that includes both improvement and destruction operators. They are able to approximately solve instances with up to 400 nodes, with respect to standard heuristics the improvements in terms of solution quality are at around 4–5 %. Silva et al. [ 33 ] propose a reinforcement learning-based model that learns eight different neighborhood functions for a Variable Neighborhood Descent heuristic with tabular Q-learning.

Exact solution methods for the CVRPTW can be divided into three categories according to [ 1 ]: Branch and Cut and Price, Branch and Cut, and reduced set partitioning. Most successful algorithms have been based on column generation, where the problem is decomposed into a restricted master problem that selects new routes from a subset of candidate routes and a pricing subproblem that generates new routes to be considered in the restricted master problem. In general, the pricing subproblem obtained is a shortest path problem with resource constraint (SPPRC) [ 34 ] which is NP-hard [ 35 ]. To obtain better lower bounds in the Branch and Bound search tree different methods have been proposed. Kohl et al. [ 36 ] proposed adding valid inequalities dynamically to strengthen linear relaxations, which results in a Branch and Cut and Price algorithm. Other families of inequalities were subsequently proposed over the years [ 37 – 39 ]. Other approaches aim at strengthening the pricing subproblem by using algorithms to generate new routes which include labeling algorithms [ 34 , 40 , 41 ] and heuristics [ 2 , 42 ], but solving the SPPRC exactly remains an obstacle. Baldacci et al. [ 43 ] therefore developed a different relaxation approach for the pricing subproblem, where a set partitioning formulation is used in which routes are dynamically generated via column generation. Their algorithm can be seen as a three-step method, where calculated lower and upper bounds are used to enumerate a subset of all feasible routes whose reduced cost with respect to the dual solution is less than or equal to the gap between the lower and upper bound. Afterwards the CVRPTW is formulated as a mixed integer program (MIP), where the previously determined subset of feasible solutions is incorporated, and solved using a commercial MIP solver. For an extensive review of all exact solution approaches, we refer to the surveys of [ 44 ] and [ 1 ].

To the best of our knowledge, the combination of quadratic unconstrained binary optimization with techniques from deep learning to solve combinatorial problems has not yet been proposed. Just recently, [ 45 ] and [ 46 ] proposed unconstrained binary models for variants of the TSP. We refer to [ 47 ] for an overview of quadratic unconstrained binary optimization for routing problems. Bengio et al. [ 24 ] present a recent survey on combinatorial optimization with machine learning techniques.

3. Problem setting and model

The purpose of this chapter is to provide a comprehensive introduction to the CVRPTW, along with a detailed explanation of the design of the deep neural network. Furthermore, we present two distinct methods for constructing solutions and elaborate on how the information obtained from the network is utilized in both approaches.

3.1. Problem setting

The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) is an extension of the classical and best known routing problem, the Traveling Salesman Problem (TSP). Given a fleet of K vehicles, the goal is to find routes, such that all nodes are visited and the capacity and time window constraints are met.

More specifically, a CVRPTW instance is given as a directed fully connected graph G = ( V, E ) with n +1 nodes, where node 0 is the special depot node. The cost for edge e = ( i, j ) is given by c ij and represents the transit costs from node i to j . Besides its coordinates, each node has additional attributes, namely a demand and a time window, imposing conditional constraints on the nodes. Therefore, we can define a problem instance of the CVRPTW to consist of:

• X = { x 1 , …, x n }, where x i ∈ [ 0 , 1 ] 2 are the coordinates of node i in the two-dimensional unit square.

• The location of the depot, given as x 0 ∈ [ 0 , 1 ] 2 .

• The demands at each node i ∈[ n ], given as D = { d 1 , …, d n }.

• T = {[ a 0 , b 0 ], [ a 1 , b 1 ], …, [ a n , b n ]}, where [ a i , b i ] are the time windows for each node i ∈[ n ] and [ a 0 , b 0 ] represents the planning horizon regarding earliest possible departure from and latest possible return to the depot.

• The capacity of the vehicles C .

Moreover each node i ∈ V requires a specific service duration h i . The aim is to find routes r k , k ∈[ K ], such that all nodes are visited. A tour r k is a sequence of nodes, starting and ending at the depot node 0, representing the order in which vehicle k visits the nodes. A set of tours R = { r 1 , …, r K } is considered a solution, if ∪ k ∈[ K ] r k = V , ∩ k ∈[ K ] r k = {0} and all tours satisfy the capacity and time window constraints. The capacity constraint is given as ∑ i ∈ r k d i ≤ C and the time window constraint states, that the time service starts at node i , s i , has to satisfy a i ≤ s i ≤ b i . An arrival at node i before a i is considered valid, the vehicle then has to wait until a i to start the service.

The components of the instances contain a few assumptions. We assume a homogenous fleet of K vehicles, so that the capacity and travel time is equal for all vehicles. Furthermore, the edge weights c ij represent the transit costs from node i to node j and without loss of generality include the service durations h i of node i .

There are different approaches to formulate the objective function. The classical objective function is to minimize the total distance traveled over all vehicles. But there are more advanced formulations of the objective, for example [ 9 ] take a more holistic perspective by also including the waiting times into the objective, searching for a good trade-off between total distance, waiting times and number of vehicles. On the other hand, especially in the operations research literature, the main objective is to minimize the number of vehicles, including the total distance just as a secondary, which has its source in cost reduction being the main focus. Other objectives include minimizing the total distance traveled while using all K vehicles (see [ 48 ]). In this work, we focus on the classical approach by minimizing the total distance, but pursuing other objectives would only need small changes in our models.

3.2. Graph convolutional network

For our model we use a graph neural network called Residual Gated Graph ConvNet (GCN) (see [ 28 ]), which was adapted for the TSP in [ 11 ]. They provide a framework to solve routing problems using the GCN, however, they only adapt it to solve the TSP with no consideration of time windows. In this section, we present our extension to address the CVRPTW, which relies on modifying the layers to adapt additional constraints within the framework and modify the search method for constructing full solutions.

The neural network outputs probabilities over the edges of the graph in order to predict which edges are most promising to be included in a solution. Complete solutions are obtained by converting these probabilities received from the model to valid tours via beam search [ 25 ], straightforward heuristics or quantum-inspired computing.

3.2.1. Input layer

The input for the node features is five-dimensional. For node i we have the two-dimensional coordinates x i ∈ [ 0 , 1 ] 2 , the time window given as [ a i , b i ] and the normalized demand d i / C , where we set d 0 = 0 for the depot. These features are concatenated to the five-dimensional input feature vector y i and are then embedded to a h 2 -dimensional representation , where h denotes the hidden dimension of our network. Similar to [ 49 ], the special depot node gets a separate learned initial embedding parameter. For that, define y ^ 0 ∈ { 0 , 1 } n + 1 to be the unit vector with entry one at the first position and zeros otherwise. This is put together as the node input feature as follows:

where A 1 ∈ ℝ h 2 × 5 , A 2 ∈ ℝ h 2 × ( n + 1 ) and ·⊕· is the concatenation operator.

For the input edge feature, the edge values c ij are embedded as a h 2 -dimensional feature vector. We do not integrate the K-nearest neighbor feature used in [ 11 ], since, in contrast to the TSP without time windows, the assumption that a node in the solution is usually connected to nodes in its close proximity (see [ 11 ]) does not necessarily hold with time window constraints. Instead, we use an indicator function δ ij of an edge which has the value one for edges connecting nodes i and j , with i ≠ j and i, j not the depot, and value two for edges connecting nodes with itself. To tag the depot as a special node, the indicator function δ ij furthermore has value 3 for edges to and from the depot and value 4 for the depot self-loop. Together, the edge input feature is given as:

where A 3 ∈ ℝ h 2 × 1 and A 4 ∈ ℝ h 2 × 4 . As for the parameters A 2 , we apply a separate embedding layer to learn the embedding parameters A 4 for our indicator function δ ij .

3.2.2. Graph convolution layer

In each of the Graph Convolution layers the model updates the edge and node embeddings. Following [ 11 ], we leverage the design of the Residual Gated Graph ConvNet developed in [ 28 ] by adding an edge feature representation. Let ℓ be the current layer and for node i and edge ( i, j ), let x i ℓ be the node features vector and e i j ℓ the edge features vector. We define the features for layer ℓ+1 in the following way:

where W k ∈ ℝ h × h for k ∈[5], σ is the sigmoid function, ε is a small value, ReLU being the rectified linear unit, BN stands for batch normalization, N ( i ) denotes the neighborhood of node i , ·⊙· denotes the Hadamard product operator and η i j ℓ being defined as

For the input layer, we set x i 0 = α i and e i j 0 = β i j . We implement W 5 ℓ as a separate parameter in order to allow the model to distinguish different directions of edges, since in the context of CVRPTW we have directed edges in our solutions. The training labels for the edges are also set accordingly, meaning if edge ( i, j ) is contained in the solution, then edge ( j, i ) will have label zero, although edge ( i, j ) is labeled with a one (see [ 49 ]). Batch normalization is a mechanism that normalizes layer inputs in order to reduce internal covariate shifts which allows the usage of higher learning rates and hence accelerates the learning of deep architectures (see [ 50 ] for more details).

3.2.3. MLP classifier

A Multi-layer Perceptron (MLP), which is a fully connected feedforward neural network with a number ℓ C of hidden layers, is used for generating the desired output, a finite measure that represents probabilities over the edges of our fully connected graph. For each edge embedding e i j L of the last Graph Convolution layer L , the MLP outputs the probability p ij that this edge is included in the tours of the CVRPTW solution:

The edge representations are linked to the ground-truth tour through a softmax output layer, which allows us to train the model parameters end-to-end by minimizing the cross-entropy loss via gradient descent (see [ 11 ]).

In the following sections, we describe two methods which use these edge probabilities to build valid solutions.

3.3. Beam search

To create valid solutions from our network model's output, we cannot simply select the edges with the highest probability until all nodes are visited, as this often results in invalid tours. Instead, we use beam search [ 25 ], a limited-width breadth-first tree search, to construct solutions.

Starting from the root node (which may be the depot but can also contain an initial partial solution), in each layer of the search tree, only a subset of the nodes with regard to a scoring policy are further explored. The descendants of a node i in layer ℓ are those nodes that are eligible as the next stop for the partially constructed tour represented in i . In the context of CVRPTW, where we have dynamic parts of partial solutions, such as the current point of time and the already occupied capacity of the vehicle, we apply a masking strategy to efficiently build valid solutions. This is done by masking out invalid descendants in layer ℓ+1 with respect to the time window and demand constraints as well as the already visited nodes in this partial solution. Then, from the set of all nodes in layer ℓ+1, only a subset containing the b best nodes (with respect to the scoring function) are retained, the other nodes are discarded. The parameter b is called the beam width. In our case, the scoring function are the probabilities gained by the GCN and we choose the b nodes whose connecting edges hold the highest probability. This is done iteratively, until all nodes in the graph are visited. If a node contains a full solution, the solution is evaluated and stored. The beam search stops when no more branches are possible on the current level, i.e., when b complete solutions have been found. The final solution then is the one which yields the highest score with respect to the scoring function, which translates to having the highest probability out of the b found solutions regarding the output of our neural network.

Beam search is asymptotically optimal for b = n ·2 n , but choosing a smaller b allows us to trade quality for computational performance and memory needed, since it decreases the search space but possibly the best solution is pruned. Furthermore, the beam search can take a sparse graph instead of a fully connected graph as an input to accelerate the tree search. This enables us to utilize the neural network in a second manner, as we can set a threshold to the edge probability given by the neural network for an edge to be included in the sparse graph, which is then given as input for the beam search. This can be interpreted as a learned problem reduction [ 12 ]. We use a low threshold of at least 10 −4 , which already excludes most of the edges. For details about the reduction see Section 4.3.1.

The approach of choosing the solution with the highest probability produced by the beam search is called GCNBS in the following. However, out of the b solutions found, we can also select the one having the overall shortest tour. This follows the approach in [ 17 ], where they sample a set of solutions and select the shortest one as the final solution out of this set, and can be interpreted as a shortest tour heuristic [ 11 ], which is therefore called GCNBSSTH in the rest of the paper.

3.4. Quadratic unconstrained binary optimization

In this section, we present a second approach to build feasible solutions to the CVRPTW using our proposed neural network model. In recent years, the development in quantum technologies led to bigger interest in formulating combinatorial optimization as quadratic unconstrained binary optimization problems (QUBO), as these formulations are suited best to be solved via quantum computing (see [ 47 ]). The CVRPTW imposes time window constraints, which are expressed as inequality constraints. Generally speaking, inequalities are known to be particularly difficult to handle within QUBO models (see [ 45 ]).

Building on the work of [ 46 ], we develop a new formulation for the CVRPTW as an quadratic unconstrained binary optimization problem. We work with an integer linear programming (ILP) formulation based on edge presentations as a starting point, as this approach is best suited to benefit from our learning-based problem reduction presented in Section 3.2. We derive a novel QUBO formulation for the CVRPTW and asymptotically calculate the number of variables needed to model this formulation. From this QUBO formulation, we derive a binary quadratic program (BQP), which is then used to solve the CVRPTW on hardware specialized for solving BQPs and QUBO problems. We use the BQP because it allows us to solve larger problem instances on said hardware. This is explained in detail in Section 3.4.5.

3.4.1. Introduction to QUBO

Oftentimes, optimization problems can be formulated as finding the minimum of a function f which models problem p . The global minimum of f represents the optimal solution of p . Quadratic unconstrained binary optimization aims at formulating optimization problems as quadratic polynomials, where the decision variables are binary. In detail, a QUBO problem is of the form

where x is our decision vector containing the binary decision variables and Q is a square upper-triangular matrix taking values in the reals. The goal is to find the vector x * that minimizes f . This general form includes quadratic as well as linear objective functions, if Q is a diagonal matrix and one notices that x i 2 = x i for x i ∈{0, 1}.

A more comprehensive introduction into QUBO can be found in [ 51 ].

3.4.2. General approach from ILP to QUBO

In general, given an ILP problem of the form

where a i , b ∈ℝ, we can obtain a QUBO formulation by transforming the equality constraints into the objective function:

obtaining a new function which is equal to the original ILP if and only if the binary decision variables z i fulfill the equality constraint. The value for the penalty constant P ∈ℝ ≥0 has to be set to weigh the constraints. Now, lets assume we are given an ILP problem, where the variables z i are not binary, but rather integer variables with bounds z i ℓ ≤ z i ≤ z i u . Since a QUBO problem is only able to handle binary variables, we have to convert the integer variables to binary by replacing each z with its binary expansion:

where k z : = ⌈ log 2 ( z u − z l + 1 ) ⌉ and x z, j are new binary variables. If linear inequality constraints are given of the form

they must be converted into equality constraints by adding slack variables λ to derive ∑ i = 1 k a i z i + λ = b . These additional integer slack variables also have to be optimized, resulting in a larger representation of the original problem after applying the binary expansion (9). In order to bound the number of additional variables needed, we can define sharp upper and lower limits for the value the slack variables can take. Generally speaking, it holds that

but problem specific knowledge oftentimes allows to find sharper bounds.

3.4.3. ILP formulation of the CVRPTW

Given the directed graph G = ( V, E ), with V = {0, …, n } and 0 being the depot, and a fleet of K vehicles each with a capacity of C , let the decision variables x i, j for i, j ∈ V , with i ≠ j , be defined as

Note that in order to minimize the number of decision variables needed, the point in time in which the edge is used is not specified with the decision variable. To model the time window constraints, let the variable s i represent the time at which the vehicle arrives at node i . Each node i ∈[ n ] has an associated time window [ a i , b i ] and demand d i and the edge costs representing the travel time between nodes is given as c ij for edge ( i, j ). Let y i be the available capacity of the vehicle after visiting node i ∈[ n ]. The ILP for the CVRPTW can be formulated as follows:

where (13) are the assignment constraints requiring each customer to be served by exactly one vehicle (note that the depot is excepted), (14) and (15) are the flow constraints, (16) and (17) are the capacity constraints, where (16) guarantees the demands at each nodes are loaded and (17) restricts the maximal load to the capacity of the vehicle. (18) linearizes the conditional statement that, if edge ( i, j ) is used, then the arrival time at node j is at least the arrival time at node i plus the cost to get from i to j . The constant M can be set to max i , j { b i + c i , j - a j } (see [ 52 ]). Constraint (19) guarantees the arrival time to be within the time window, while constraint (20) sets the maximum number of vehicles to be used. Finally, (21) and (22) are the binary and integer constraints. Before we start transforming the ILP problem to a QUBO formulation, we can simplify this ILP formulation in order to minimize the number of variables needed. We are able to remove the inequalities stated in constraint (17) and (19), since the upper and lower bounds for the integer variables are already used while converting those integer variables to binary and therefore are not explicitly needed in the formulation. Note that we do not need to include subtour elimination constraints, as the time window constraints impose a unique route direction and therefore eliminate any subtours (see [ 52 ]).

A more natural way to formulate the CVRPTW as an ILP is to define the decision variables as

With the decision variables being structured by also having index k for specific vehicles, one could keep track of the capacity constraints by simply adding the inequality

for each vehicle k ∈[ K ], which eliminates the need for the additional integer variables y i . By using this formulation, the number of inequalities needed to model the CVRPTW would decrease significantly. Specifically, constraint (23) would add only K inequalities, instead of the n 2 inequalities required for our formulation of the capacity constraint (16). This reduction in the number of inequalities also leads to a decrease in the number of slack variables needed to reformulate all inequalities to equalities. The final QUBO formulation requires that both integer variables and slack variables are represented in binary form. As a result, the number of variables in the final formulation increases for each additional integer and slack variable. But on the other hand the number of decision variables x would increase to n 2 K . Since we utilize Fujitsu's Digital Annealer [ 53 ] to solve our QUBO formulation, which can handle inequalities without additional slack variables (see Section 3.4.5), we found that our current formulation, that uses a smaller number of decision variables and more inequality constraints, is better suited for our computational experiments, as the higher number of inequalities is comparatively insignificant to the number of decision variables.

3.4.4. QUBO formulation of the CVRPTW

In order to transform this ILP model into a QUBO formulation, we have to change the inequalities to equalities by introducing slack variables. To convert those slack variables into binary variables we have to define upper bounds for each slack variable. Let us start with slack variable λ i , j 16 for inequality (16). Formula (10) gives us

For the slack variable λ i , j 18 for constraint (18) we have

The slack variable for constraint (20) clearly can be bounded by λ 20 ≤ K . Now we are able to state the full quadratic binary polynomial f Q modeling the CVRPTW. Following [ 46 ], for simplicity we state the function including all constraints including those which hold a i + c ij > b j . Applying the binary expansion function B , which is defined in (9), with the defined upper bounds for our integer variables and slack variables, the function can be stated as

The penalty constants P i ∈ℝ ≥0 , i ∈[4], have to be adjusted accordingly, such that a violation of the constraints in f route , f cap or f tw results in a larger increase of the function value than the decrease it might produces in the value of f obj . Selecting penalty constants for the constraints greater than the optimal tour length provides a theoretical assurance that the QUBO solution with the lowest energy corresponds to a feasible solution. Computational experiments have shown that the performance from the QUBO solver we applied is best when choosing the penalty constants sufficiently large, but not arbitrarily large. Fujitsu's Digital Annealer offers the functionality to automatically adjust the penalty coefficients during the optimization process, a description of that process is found in Section 3.4.5.

Utilizing the lower and upper bounds of integer variables for the binary expansions allows the prevention of excessively large coefficients for the newly introduced binary variables in the majority of cases. But using these binary expansions is still problematic as it can cause difficulties for solvers to find the correct assignments [ 54 ]. For example, if an integer variable's value needs to be changed from 16 (binary encoding: 10000) to 15 (binary encoding: 01111) during the optimization process, five bit switches are required. This becomes increasingly more challenging as the values of the integer variables increase, as it leads to large coefficients for the binary variables, further complicating the process of finding the correct assignment for each binary variable. In Section 4, we will investigate the impact of an increasing number of binary expanded variables on the solver's ability to identify solutions.

We are now able to determine the number of variables required for this formulation. For the binary representation of the integer and slack variables, let us look at equation (9) again. For each integer variable z , the number of new binary variables added to the model is exactly k z : = ⌈ log 2 ( z u − z l + 1 ) ⌉ . The integer variables s i , y i and slack variables λ i , j 16 , λ i , j 18 , λ 20 therefore require at most ⌈ log 2 ( max i b i - a i + 1 ) ⌉ , ⌈log 2 ( Q +1)⌉, ⌈ log 2 ( max i 2 Q + d i ) ⌉ , ⌈ log 2 ( max i , j - a i + b j + M ) ⌉ and ⌈log 2 ( K +1)⌉ binary variables, respectively. If we define δ to be

then for every integer and slack variable at most δ binary variables are required for the binary encoding. Overall, we need O ( n 2 ) variables to represent the x i, j . For the integer variables s i , y i we have O ( n δ ) variables for the binary encoding. Since we have O ( n 2 ) inequalities, we need O ( n 2 δ ) additional slack variables in binary form. Thus, in total O ( n 2 + n 2 δ ) variables are required for our QUBO formulation of the CVRPTW.

As pointed out earlier, some variables can be removed beforehand in order to lower the total number of variables needed. This holds for example if a i + c ij > b j for some i, j ∈ V or we do not have a fully connected graph to begin with. In the first case we can simply set x i, j = 0. In the latter case, the overall number of variables needed is reduced to O ( | E | 2 + | E | 2 δ ) .

3.4.5. Fujitsu's Digital Annealer

We use Fujitsu's Digital Annealer (DA) [ 53 ] to solve the QUBO formulations presented in Section 3.4.4. We call this approach GCNDA. The DA is a product developed by Fujitsu to fill in the performance gap between classical computers, which hit their limit rather quickly when solving larger QUBO formulations, and quantum computers, which are still in its experimental stage. The DA is a hardware system built by Fujitsu specialized on finding the minimum of binary quadratic polynomials by using parallel computation.

The Digital Annealer in its 3 rd generation is able to handle optimization problems with up to 100,000 decision variables, a major improvement from the 8192 supported decision variables in its 2 nd generation. This is done by joining multiple Digital Annealer Units (DAU) together. These DAUs are dedicated processors executing the minimization algorithm parallel tempering and can be seen as 2 nd generation DAs. Unlike the 2 nd generation DA, the 3 rd generation DA cannot solve QUBO formulations with up to its maximum number of decision variables fully on the dedicated processor. Using additional software, the large QUBO formulation is decomposed into smaller QUBO formulations, which are then minimized on one or multiple DAUs, depending on the problem size.

Another major improvement from its 2 nd to 3 rd generation is the ability to handle inequalities. As shown in Section 3.4.4, formulating inequality constraints as binary quadratic polynomials on the one hand is connected to a lot of work, as one needs to convert those inequalities to equalities by adding slack variables. To limit the number of variables needed for the formulation, finding sharp upper and lower bounds for those slack variables is important. On the other hand, adding those slack variables and subsequently converting them to binary variables adds a large number of decision variables to the formulation, which makes it harder to solve. All this is obsolete when using the DA, as one can declare the inequalities separately from the QUBO formulation and is therefore able to solve optimization problems in the form of binary quadratic programming problems (BQP). This means that inequality constraints are not converted to a penalty term and thus do not use any decision variables. Using this feature, we can remove f cap and f tw from the energy function in (24), leaving us with a substantially smaller BQP. A detailed experimental analysis of the size of the problem formulation with and without inequalities is done in Section 4.3.1. There is no sharp upper bound for the number of inequalities the DA is able to handle, but it is limited to a number in the lower 6-digits range. It depends on the complexity of the BQP formulation to solve, which has to be examined experimentally.

The Digital Annealer has additional features such as Auto-Scaling, which automatically scales the penalty factors in the energy function (24). For that, the minimization and penalty terms are passed separately to allow the adjustment of the penalty coefficient P penalty . Starting from an preset initial value, P penalty is multiplied with an increase value in [1, 2] every time the objective value is not improved for a set number of iterations. For a more detailed explanation including examples concerning the Digital Annealer for fast combinatorial optimization we refer to [ 55 ].

4. Computational experiments

We evaluate our approaches on three different problem sizes, ranging from 20 to 100 nodes. For each problem size we train our neural network on different datasets. We report the results of our approaches and compare them to highly developed heuristics like LKH3 [ 14 ] and Google's OR-Tools [ 15 ], which frequently serve as heuristic baselines in related work, as well as the commercial state-of-the-art exact solver Gurobi [ 13 ]. The heuristic LKH3 plays a crucial role in this, as it serves as both the source of training data for our neural network and the benchmark against which the quality of solutions generated by other methods is evaluated. We refer to this as the solution gap, which denotes the disparity between the solutions produced by various methods and the solution obtained by LKH3. The found solutions are compared in terms of costs, solution gap, and computation times. We show that our approaches achieve results close to the LKH3 solutions on smaller to medium sized problem instances, and outperform Gurobi on large instances achieving better results with less computation time.

4.1. Implementation and hyperparameters

All models are implemented in Python 3.9 and run under Linux. The neural network architecture is implemented using PyTorch version 1.12.1 [ 56 ]) to use GPU computation with Cuda version 11.3.

Our neural network model does not contain a large number of hyperparameters. The graph convolutional neural network consists of ℓ GCN = 30 hidden layers and ℓ C = 3 layers in the MLP. We use a hidden dimension h = 300 in each of the layers. For the beam width b we use different values from b = 1, 000 to b = 10, 000. The threshold for edges to be included in the sparse graph is set to 10 −5 for GCNBS and GCNBSSTH and ranges from 10 −3 to 10 −5 for GCNDA.

For Fujitsu's Digital Annealer we use the following parameters: In the energy function (24), P 1 is set to one and the penalties P 2 , P 3 , and P 4 are automatically incrementally adapted to the optimal value. This is done by multiplying them by 1.5, if the result did not improve for 500 iterations. That way, the Digital Annealer focuses on satisfying the side constraints before minimizing the objective f obj . The time limit for the optimization process is set to 1 s per two bits. The selection of the parameter values for the automatic adaption is based on empirical experiments, which demonstrate that the parameters provide a good balance between the performance of the DA and the computational resources required by the optimization process. This amounts on average to a time limit of roughly 300 s for instances with 20 nodes, 800 s for the 50 nodes instances and 1,700 s for instances with 100 nodes, as can be seen in Table 2 . Given the substantial differences in underlying technology, making a direct comparison to CPU times is difficult.

For the implementation of the beam search, we define similar to [ 49 ] an auxiliary graph G ′ = ( V ′, E ′), which is obtained from the input graph G = ( V, E ) for the beam search (either fully connected or sparse) in the following way: For V = {1, …, n } with 1 being the depot node, let V ′ = (1, …, n, n +1, …, 2 n −1) be the set of nodes, where we add for each node i in the original graph (except the depot) a new node i ′. Connections to these new nodes denote connections via the depot to node i . This allows us to implement the beam search such that at step k in the beam search exactly k of the n nodes are visited, so that comparisons of partial solutions and building the full solution incrementally are more straightforward. The transition probabilities c ij for edges ( i, j ) for i, j ∈{1, …, n } are given by the neural network. For edges ( i, j ′) with j ′∈{ n +1, …, 2 n −1} we follow [ 49 ] and set c i j ′ = c i 1 · c 1 j · 0 . 1 , where multiplication is used so that c i j ′ ∈ ( 0 , 1 ) can still be interpreted as a probability. The factor 0.1 is multiplied to incentivize building as few routes as possible and therefore implicitly minimize the number of vehicles used.

Remember that infeasible connections concerning the demand and time window constraints are masked out in each beam search step, so that a connection from i to a node j ′∈{ n +1, …, 2 n −1} is only included if the travel times from i via the depot to j = ( j ′ mod n −1) are feasible with regard to the time window constraints. On the other hand, if a connection from i to a node j ∈{1, …, n } is masked out for a partial solution because of the demand constraint, the connection from i to j ′ = j + n −1 is included, as the route via the depot resets the occupied capacity.

4.2. Experiment setup

4.2.1. data generation.

We sample problem instances based on the distribution given in the R201 instance of the benchmark set of [ 57 ], which consists of randomly generated geographical data, a long scheduling horizon and short to medium sized time windows allowing only a few customers per route. We follow the approach used in [ 9 ] for the data generation.

For the instances, the locations of the nodes are sampled uniformly random in the square [0, 100] 2 . We set capacities to C 20 = 500, C 50 = 750 and C 100 = 1, 000 with respect to the problem sizes. We choose to sample the demands d i for the nodes according to the R201 distribution from a normal distribution q ^ ~ N ( 15 , 10 ) , as for the R201 instance the demands have a mean of 17.24 and standard deviation of 9.4175, and then rounding down the values to integers. The time windows are sampled such that they are feasible with respect to the travel time needed from the depot to the node. This is done by defining a suitable horizon h i = [ h i 0 , h i 1 ] for each node i depending on the distance from the depot to node i , sampling the start a i of the time window uniformly from h i and then sampling the end b i uniformly from the interval [ a i ^ , h i 1 ] , where a i ^ = a i + 300 ε with ε = max ( | ε ^ | , 1 / 100 ) , ε ^ ~ N ( 0 , 1 ) .

To generate solutions for the randomly sampled instances, we use the High Performance Computing Cluster available at Hamburg University of Technology, and heuristically solve each instance using one run of LKH3 [ 14 ] on machines with two CPUs of type Intel Xeon E5-2680v3 @ 2,50GHz with 12 Cores.

4.2.2. Training and evaluation

For each problem size an individual neural network is trained on 1 million instances with the respective number of nodes, which is split into training, test, and validation sets with a 80/10/10 ratio. We apply a supervised learning procedure, where, given as input a graph with the additional node features time windows and demands, the model is trained to output a probability matrix by minimizing the cross-entropy loss via gradient descent with respect to the adjacency matrix corresponding to the target solution. We utilize the Adam optimizer [ 58 ] along with a gradual decrease in the learning rate for smoother convergence, starting at a rate of 10 −3 . The target solutions are generated using the LKH3 heuristic and are therefore not necessarily optimal. We train using a batch size of 24, which was chosen as the result of manual tests on our GPUs. We train the nets for 1,500 epochs with 500 randomly chosen batches and select the point of training with the lowest validation loss. The training procedure is executed on machines with two CPUs of type Intel Xeon E5-2680v3 @ 2,50GHz with 12 Cores and four NVidia Tesla K80 GPUs with 12GB RAM each. We note however that it is not necessary to use a multi-GPU setup to train or evaluate our models, identical results can be attained by training longer on a single GPU.

For the evaluation of our model GCNBS, we use the same machines as for the training procedure. The preprocessing for the model GCNDA is done on a machine with an Intel Core i7-8650U CPU @ 1.90GHz and one Nvidia Geforce GTX 1050 with 2 GB RAM, the optimization is executed on Fujitsu's Digital Annealer Hardware. The baseline models are run on a machine with two CPUs of type Intel Xeon E5-2680v3 @ 2,50 GHz with 12 Cores.

4.2.3. Baseline models

We compare our models to the commercial exact solver Gurobi version 10.0.0 [ 13 ], as well as the highly-optimized heuristic solvers LKH3 [ 14 ] and the Google OR-Tools (GORT) version 9.5.2237 library [ 15 ], both of which frequently are used as heuristic baselines in related work. Although Gurobi may not be the best exact solution method, it serves as a representative example of commercial solvers. Gurobi is applied to the two-index ILP presented in Section 3.4.3 and solves the model with a single run using all available threads. Since Gurobi strives for finding the best solution, we have to set a time limit after which Gurobi stops searching for better solutions. We set this time limit to 1,800 s, a lower time limit does not yield any high quality solutions for large instances. But this only allows us to use Gurobi on the Solomon benchmark sets [ 57 ], as the 1,000 instances test sets would need too long to compute. LKH3 is executed using one run per instance. For GORT, one can choose different configurations regarding the underlying local search heuristic. Experiments have shown the guided local search (GLS) to be best suited for our needs. A time limit of 30 s per instance is set for GORT.

4.3. Results

We compare our results in terms of quality of the solutions and computation time with results from commercial solvers (Gurobi) and heuristics (LKH3, GORT). However, the calculation times are generally difficult to compare, as they depend heavily on different factors such as the implementation (Python vs. C++) and which hardware (GPUs vs. CPUs) was used. Therefore, the comparison of the results and run times can only be done conditionally.

For most algorithms it holds that a trade-off between run time and performance is possible. In our model GCNBS this means choosing the beam width, whereas for the GCNDA the choice of the time limit is the decisive factor. Instead of performing a full analysis and report of the pareto efficiency, we selected values for the beam width based on related literature, reporting the different results in order to provide an overview of the possible trade-off between run time and performance.

4.3.1. Sparsity of graphs

Our goal is to make a specific problem instance smaller by using a threshold on the edge probabilities that are generated by our neural network. This threshold should be set in a way that it allows us to run the GCNBS more efficiently, while also reducing the size of the instance enough to run the GCNDA on the Digital Annealer. Therefore, we first have to evaluate the trade-off between the value of the probability threshold and the number of reduced edges and the resulting instance size.

Table 1 shows the mean number of deleted edges over 10,000 instances for the different problem sizes and edge probability thresholds. One can see that even with a high threshold of 0.001, the neural network confidently excludes most of the edges. The percentaged number of deleted edges for the same threshold increases with the problem sizes, since for larger graphs the percentage of edges irrelevant for the solution increases. A threshold of 0.00001 excludes at least 3/4 of the edges, for graphs with 100 nodes the neural network is even able to exclude all but 10 percent of the edges, which reduces the number of edges included in the graph from 10,201 to just under 1,000 edges on average.

Table 1 . Number and percentage of edges deleted for different problem sizes and thresholds (mean over 10,000 instances).

For the GCNDA, these sparse graphs act as a learned problem reduction. Table 2 analyzes the impact of the problem reduction on the size of the instances for the Digital Annealer. For the number of binary variables needed to represent integer variables, for our datasets an upper bound for δ in equation (25) is given as δ = 10. Note however that we can reduce the number of auxiliary binary variables needed by taking into account the upper and lower bound of each integer variable individually before initializing the auxiliary binary variables. As shown in Table 2 , the learned problem reduction allows us to reduce the size of our instances, consisting of the QUBO formulation and the inequalities, significantly.

Table 2 . Estimation of instance sizes for Digital Annealer after applying learned problem reduction on 10,000 instances.

4.3.2. Experimental results

Tables 3 , 4 show the results of our models for the three different problem sizes with beam width b = 1, 000 and b = 10, 000, denoted as 1K and 10K in the model name, for two different datasets and compares our results to results obtained by the previously described exact and heuristic baseline solvers. We report the results for two datasets, the first consists of 1,000 randomly sampled instances from the distribution described in Section 4.2.1. The second dataset is the well-known Solomon benchmark set, which consists of 56 instances with 100 customers. For the smaller problem sizes, we split each instance with 100 customers in two and five instances respectively, with the depot being the same from the original instance. We report the average gap to the groundtruth solution, which was obtained by LKH3, as well as the average calculation time per instance in seconds. As Gurobi was not able to find solutions for all 46 instances of the problem set with n = 50 nodes, the mean cost in Table 5 is not comparable to the other approaches. Even with a time limit of 7,200 s per instance, Gurobi managed to find solutions only for 37 of 46 instances. According to the data presented in Table 4 , for instances with n = 20 nodes, Gurobi, an exact solver, exhibited a gap with respect to the best known solutions, although generally not reaching the time limit. This finding can be attributed to Gurobi's inability to identify the globally optimal solutions for all 115 instances of the dataset within the imposed time constraints, despite not reaching the limit for the majority of the cases.

Table 3 . Mean cost, gap, and time per instance to solve 1,000 CVRPTW instances.

Table 4 . Mean cost, gap, and time per instance to solve the Solomon CVRPTW benchmark set.

Table 5 . Mean cost, gap, and time per instance to solve the Solomon CVRPTW benchmark set R.

In general, our learning-based approach is able to improve its performance over the initial setting by additionally applying the shortest tour heuristic without noticeable implications regarding the computation time. It is evident that both approaches, GCNBS and GCNBSSTH, yield better results for all instance sizes and beam widths for the 1,000 instances dataset in comparison to Solomon's benchmark set [ 57 ], which can be attributed to the fact that the neural networks were trained with datasets based on Solomon's benchmark set R [ 57 ]. This demonstrates that the performance of the proposed approach is heavily reliant on the selection of training data. In order to achieve generalization within the same problem class, the model must be trained with data that is representative of the underlying distribution, which can vary considerably across different applications. The process of generating suitable training data is a challenging but critical task, as the efficacy of a supervised learning approach is directly influenced by the quality of the data utilized for training.

For small instance sizes, with a beam width of 10,000 GCNBSSTH finds solutions close to the LKH3 results with a mean gap of approximately 1.5% on the dataset used to also train the neural nets and around 2.8% on the Solomon dataset. GCNBS (+STH) outperforms the commercial solver Gurobi [ 13 ] in terms of speed for all instance sizes. In fact Gurobi was only able to find solutions for instances with 20 nodes reliably within a time limit of 1,800 s. For instances with 50 nodes, Gurobi was only able to find a valid solution approximately half of the time, whereas for instances with 100 nodes, Gurobi could not find valid solutions within the time limit. Within a setting of similar computation times, the assumption is that similar results regarding solution quality can be expected for instances with 20 nodes by our learning-based approach if trained for the specific dataset distribution. The highly optimized heuristic LKH3 [ 14 ] as well as Google's OR-Tools [ 15 ] outperformed GCNBSSTH in terms of speed and solution quality on all instance sizes. For smaller instances, Google's OR-Tools perform even better than LKH3, whereas for instances with 100 nodes LKH3 shows the best results. Especially for large instances with 100 nodes the shortcoming of our model becomes apparent, as the beam search becomes computationally expensive for larger instances and beam widths. The computation time mainly depends on the chosen beam width and it seems to be close to a linear correlation between beam width and computation time. Tests have shown that most of the computation time in the current implementation is used in the masking of invalid next nodes with respect to the time windows constraints within the beam search and a more efficient implementation of the masking scheme could lead to significant improvements regarding the computation time.

Table 5 shows the results of our experiments on the Digital Annealer compared to results obtained by GCNBS (+STH) as well as the baseline models. As the amount of time to run experiments on the Digital Annealer was limited, we conducted the experiments on a smaller dataset only consisting of the Solomon benchmark instances from the set R, which was also the baseline distribution for our data generation. The dataset consists of 115 and 46 instances for the problem sizes with 20 and 50 customers, respectively. The asterisk on Gurobi's result for n = 50 indicates that the results are not comparable as for only half of the instances Gurobi was able to find feasible solutions.

The performance of the GCNDA method is inferior to that of GCNBSSTH and especially the baseline models, in terms of both run time and quality of results. The use of integer variables poses a significant challenge for the Digital Annealer (DA) when solving instances of the CVRPTW. As previously mentioned, each integer variable must be represented in binary form, resulting in O ( n δ ) binary variables for our model, which contains 2 n integer variables. As discussed in Section 3.4.4, the requirement to identify the correct assignment of all binary variables introduced for binary expansions poses significant challenges for QUBO and BQP solvers, as it necessitates a considerable number of bit switches. However, the results from Table 5 clearly demonstrate the effectiveness of the learned problem reduction method in improving solution quality by reducing the size of the considered graph for smaller instances. A performance improvement of around 3%, while also accelerating the optimization process, is a promising result. It is reasonable to assume that the decrease in computation time becomes more obvious if a higher time limit is given. As seen in Table 2 , the higher the edge probability threshold used in the reduction, the fewer integer variables are required. The risk of the best solutions not being obtained due to the removal of important edges during the reduction process is found to be negligible within the range of solution gaps presented in Table 5 . As the size of the CVRPTW problem increases, the optimization of the number of integer variables becomes increasingly challenging. Since the learned problem reduction only applies to the decision variables x , whereas the number of integer variables y and s is not reduced, the effect of the reduction for larger instances is negligible. Given the current version of the DA and the resources available for our experimental study, the limitations of the approach have been reached.

4.3.3. Example solution visualizations

Figures 1 , 2 show solution plots for two different instances with 20 nodes as well as the groundtruth tour in the upper left corner and the probabilistic heatmap output of our Graph Convolutional Network in the upper right corner of Figure 1 (left and middle plot in Figure 1 , respectively). A darker red color in the heatmap implies a higher probability and only edges with a probability of at least 0.25 are plotted. The gray edges show all edges of the fully connected graph. In Figure 2 , one can see the improvement of the solution by applying the shortest tour heuristic. The solution of the beam search in fact uses one vehicle less than the groundtruth tour, but by using one more vehicle, GCNBSSTH is able to lower the total distance traveled and find the best solution. Figure 2 showcases the ability of GCNBSSTH to correctly identify most of the important edges as important, while also excluding most of the irrelevant edges. But in contrast to the solution found in Figure 1 , here GCNBSSTH uses one vehicle less than the best solution, illustrating the need for more information regarding other subtours while building the solution in order to find the best solution.

Figure 1 . Example solution plot GCNBS and GCNBSSTH for n = 20.

Figure 2 . Example solution plot GCNBSSTH for n = 20.

Figure 3 visualizes the optimization process by the Digital Annealer for four different instances. The x-axis refers to the solution time and the two y-axes indicate the energy of the objective function and the constraints (penalties), respectively. The DA starts with a condition where the energy of the objective function is zero and the energy of the penalties is positive. Then the solution is optimized by first decreasing the energy of the penalties to zero, which translates to finding a valid solution, and implies increasing the energy of the objective function. Then, the energy of the objective function is decreased while the energy of the penalties remains at zero.

Figure 3 . Visualization of the optimization process by the Digital Annealer for four different instances.

5. Discussion

In this paper, we proposed two learning-based approaches to solve the capacitated vehicle routing problem with time windows. The first method combines a Graph Convolutional Network with a classical tree search heuristic, whereas in the second approach the Graph Convolutional Network is used as a learning-based reduction combined with methods from the field of quadratic unconstrained binary optimization in order to solve the problem with quantum-inspired computers.

In conclusion, the proposed approaches for solving the capacitated vehicle routing problem with time windows (CVRPTW) using a Graph Convolutional Network combined with a classical heuristic, a beam search, as well as in combination with quantum-inspired computing methods showed promising results. Although our approach is not as advanced as other existing heuristics and the computational results for the approach using a beam search were not quite as good as those of state-of-the-art highly developed heuristic approaches such as LKH3 [ 14 ] and Google's OR-Tools [ 15 ], it is still competitive on smaller instances, showing promising results close to the known best results on datasets containing instances with 20 and 50 nodes, respectively, while also outperforming the exact solver Gurobi [ 13 ] in terms of solution quality and computation time for larger instances. It is apparent from the computational results that the proposed model is less efficient compared to the highly optimized heuristics that leverage the problem structure through the use of expert knowledge. However, the objective of our approach is to demonstrate the capability of deep learning to construct models capable of solving complex combinatorial optimization problems with minimal dependence on expert knowledge, while simultaneously achieving superior performance compared to existing commercial solvers.

The binary quadratic program (BQP) formulation of the CVRPTW, which we derived from our developed novel quadratic unconstrained binary optimization (QUBO) formulation of the CVRPTW, was solved on Fujitsu's Digital Annealer by applying a deep neural network as a learned problem reduction . This demonstrates the potential of using learned problem reductions in improving solution quality and reducing computation time on quantum(-inspired) computers. However, the large size of CVRPTW instances and the requirement for a significant number of integer variables to model the problem as a quadratic unconstrained binary optimization problem make it challenging to optimize for larger instances, even with the reduction of the instance size. Further research is necessary to identify more efficient problem representations to effectively solve CVRPTW via QUBO and to handle integer variables in a more efficient way. The computational experiments conducted have highlighted a significant dependency of the proposed approach on the training data used, as evidenced by the inferior performance observed for a data set exhibiting a slightly different data distribution when compared to instances sharing the same data distribution.

Our approaches represent novel and innovative learning-based solutions for the CVRPTW and offer an alternative solution that may be suitable for certain applications. Further research is needed to improve the performance and to validate its results on a larger and more diverse set of inputs. To further develop approaches that use deep neural networks build solutions constructively is a promising direction for future research, as the constructive nature allows for efficient handling of hard constraints, such as time windows, which can be difficult for both neural combinatorial optimization and local search heuristics to effectively address. These types of constraints are particularly challenging in these methods due to the need to maintain feasibility while adapting a solution. The use of deep learning in this context represents a promising direction for future work in solving the capacitated vehicle routing problem with hard constraints, but further research is necessary to develop more sophisticated methods for constructing solutions. Moving forward, our research aims to overcome the limitations of the beam search algorithm by incorporating contextual information from partially constructed solutions to guide the selection of the next node, and integrating a deep neural network in an autoregressive manner. Our objective is to enhance the efficiency and accuracy of the search process for complex problems by leveraging these advanced techniques.

Combining classical approaches with deep learning has the potential to improve solution methods for a wide range of combinatorial optimization problems, as it is not necessary to have in-depth, specific knowledge about the structures of the problem and its solution space in order to develop approaches that can provide usable solutions that are faster and better than available commercial solvers.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

JD confirms sole responsibility for model conception and design, data generation, computational experiments, analysis and interpretation of results, and manuscript preparation.

Open Access funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Projektnummer 491268466 and the Hamburg University of Technology (TUHH) in the funding programme Open Access Publishing.

Acknowledgments

We would like to express our gratitude to the team at Fujitsu Technology Solutions, particularly Markus Kirsch, for their valuable collaboration and assistance in utilizing Fujitsu's Digital Annealer.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: deep learning, graph convolutional network, beam search, vehicle routing, time windows, quadratic unconstrained binary optimization

Citation: Dornemann J (2023) Solving the capacitated vehicle routing problem with time windows via graph convolutional network assisted tree search and quantum-inspired computing. Front. Appl. Math. Stat. 9:1155356. doi: 10.3389/fams.2023.1155356

Received: 31 January 2023; Accepted: 08 June 2023; Published: 22 June 2023.

Reviewed by:

Copyright © 2023 Dornemann. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jorin Dornemann, jorin.dornemann@tuhh.de

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On a model-free meta-heuristic approach for unconstrained optimization

• Published: 24 June 2024

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• Wei Xia 1 , 2   na1 &
• Deming He 1   na1  

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The efficacy of meta-heuristic algorithms has been demonstrated in solving unconstrained optimization problems. Inspired by the food foraging behavior of beetles, we propose herein a model-free meta-heuristic optimization algorithm, referred to as the Beetle Antennae Search-Bellwether Swarm, which incorporates the schemes of adaptive antenna fiber length and direction vector. A joint annealing-heating scheme along with the re-initialization mechanism is proposed such that each antenna fiber length could simultaneously gradually decrease or increase in different circumstances. We also propose to adapt each individual antenna fiber length based on the best smell perception direction (bellwether). The proposed schemes of adaptive antenna fiber length yield the algorithm almost insusceptible to different initial antenna fiber lengths for a relatively large range of initial antenna fiber lengths. The softmax combiner is leveraged to obtain the direction vector such that the beetle would be steered to step toward the new randomly combined direction and arrive at a candidate position. The beetle position is further updated by evaluating whether the candidate position is conducive to minimizing the objective function. The illustrative simulations demonstrate the fast initial convergence, cost-effectiveness and feasibility of the proposed algorithm.

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An Improved Beetle Antennae Search Algorithm

Enhanced beetle antenna search: a swarm intelligence algorithm

Improved optimal foraging algorithm for global optimization

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61871104.

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School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

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