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Optimization with Engineering Applications: Heuristics, Meta-Heuristics and Hyper-Heuristics

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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 4170

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Dear Colleagues,

Solution search methods for solving optimization problems can be categorized into exact, knowledge-based heuristic, and random-based meta-heuristic algorithms. Although exact search methods consistently achieve an optimal solution, such practices are not applicable to complex engineering problems that belong to the class of NP-complete/hard problems.

Over the past few years, various heuristic, meta-heuristic, and hyper-heuristic optimization algorithms have been introduced and applied to tackle the complexities of real-world engineering problems. Heuristics are high-speed application-specific methods, but they do not effectively investigate the search space. In contrast, meta-heuristics can obtain higher quality solutions due to their ability to cope with high computational complexity. As an extension to heuristics, hyper-heuristics provide agile and high-quality solutions that are capable of changing from one application variant to a new one quickly, which is important for solving real-time engineering problems.

This Special Issue aims to report the recent advances in optimization methods from theoretical, methodological, and applied viewpoints. We invite researchers and participants from academia and industry to submit their original research and review articles concerning new and existing heuristics, meta-heuristics, and hyper-heuristics for solving optimization problems in different branches of engineering.

Dr. Mohammad Shokouhifar Prof. Dr. Frank Werner Dr. Laith Abualigah Prof. Dr. Seyedali Mirjalili Guest Editors

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website . Once you are registered, click here to go to the submission form . Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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• engineering optimization
• mathematical modeling
• combinatorial optimization
• robust optimization
• exact methods
• heuristic algorithms
• meta-heuristic algorithms
• hyper-heuristic algorithms
• evolutionary computation
• swarm intelligence
• local-search algorithms
• multi-objective optimization
• constraint handling methods
• hybrid optimization methods
• fuzzy sets and systems
• machine learning
• deep learning
• graph theory and applications
• scheduling problems
• resource allocation problems
• vehicle routing problems
• big-data analytics
• signal/image processing
• manufacturing systems
• chemical processes
• Internet-of-Things

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Systems Engineering Heuristics

Author: Peter Brook

Heuristics provide a way for an established profession to pass on its accumulated wisdom. This allows practitioners and others interested in how things are done to gain insights from what has been found to work well in the past and to apply the lessons learned. Heuristics will usually take the form of short expressions in natural language. These can be memorable phrases encapsulating rules of thumb, shortcuts or "words to the wise", giving general guidelines on professional conduct or rules, advice, or guidelines on how to act under specific circumstances. Common heuristics do not summarize all there is to know, yet they can act as useful entry points for learning more. This article overviews heuristics in general as well as some of those specifically supporting Systems Engineering practice.

• 2 Historical Background
• 3 Modern Interest
• 4 Practical Use
• 5.1 Works Cited
• 5.2 Primary References
• 5.3 Additional References

Heuristics have always played an important part in the history of engineering and shaped its progress, especially before science developed to the point when it could also assist engineers.  Systems Engineering is still at a stage at which there is no sufficiently reliable scientific basis for many of the systems being built, which has triggered a renewed interest in heuristics to fill the gap. This is especially true as the practice of Systems Engineering is extended to provide solutions to inherently complex, unbounded, ill-structured, or "wicked" problems (Churchman 1967).

Using heuristics does not guarantee success under all circumstances, but usefulness of a heuristic can be maximized if the known extent of its applicability is made clear. At their best, heuristics can act as aids to decision making, value judgements, and assessments.

Heuristics have the potential to be useful in a number of  ways:

• reduce the amount of thinking (or computation) needed to make a good decision or a choice
• help in finding an acceptable solution to a problem
• identify the most important factors to focus on while addressing a complex problem
• improve the quality of decisions by drawing on best practices
• avoid repeating avoidable mistakes
• act as an entry point to wider knowledge of what has been found to work

Historical Background

Engineering first emerged as a series of skills acquired while transforming the ancient world, principally through buildings, cities, infrastructure, and machines of war. Since then, mankind has sought to codify the knowledge of "how to." Doing so allows each generation to learn from its predecessors, enabling more complex structures to be built with increasing confidence while avoiding repeated real-world failures. Setting out the aims for engineering in the 1st Century BCE, Vitruvius proposed a set of enduring principles: Strength, Utility and Beauty. He provided many examples of their applications to the fields of engineering of the time.

Vitruvius’ writings were rediscovered in the Middle Ages, forming the basis of the twin professions of architecture and engineering. Early cathedral builders encapsulated their knowledge in a small number of rules of thumb, such as: "maintain a low centre of gravity," "put 80% of the mass in the pillars," and "observe empirical ratios between cross-section and span for cross-members.". Designs were conservative, with large margins, the boundaries of which were largely unknown. Numerous excellent structures resulted, many of which have endured to this day. When the design margins were exceeded, for example out of a desire to build higher and more impressive structures, a high price could be paid, with the collapse of a roof, a tower, or even a whole building. From such failures, new empirical rules emerged. Much of this took place before the science behind the strength of materials or building secure foundations was understood. Only in recent times has computer simulation revealed the contribution towards certain failures played by dynamic effects, such as those of wind shear on tall structures.

Since then, engineering and applicable sciences have co-evolved: science providing the ability to predict and explain performance of engineered artifacts with greater assurance, and engineering developing new and more complex systems, requiring new scientific explanations and driving research agendas. In the modern era, complex and adaptive systems are being built which challenge conventional engineering sciences, with the builders turning to social and behavioral sciences, management sciences, and increasingly systems science to deal with some of the new forms of complexity involved and to guide the profession accordingly.

Koen (1985), went further, and argued that the whole of engineering was essentially heuristic in nature, since engineers could never know everything about whether what they built would meet their original intent, and how it would interact with an uncertain world. According to Koen, we must do what we can to tie these things down but make allowances for what we cannot know. He defined the engineering method as: the strategy for causing the best change in poorly understood or uncertain situation within available resources . He further claimed that heuristics are the only way of guiding our actions in these circumstances, and that the set of all heuristics agreed or used by a profession at any one time represents its ‘state of the art’. He also put forward a number of universal heuristic principles for the conduct of engineering, covering areas such as: rules of thumb, factors of safety, engineers’ attitude towards their work, managing risk and allocating resources. Although much of what Koen (a chemical engineer by profession) had to say remains relevant to systems engineers today, there is renewed optimism that scientific SE principles can be uncovered to underpin the discipline, just as they have done through the ages. (see Systems Engineering Principles )

Modern Interest

Renewed interest in the application of heuristics to the field of Systems Engineering stems from the seminal work of Rechtin and Maier (2009). Their book remains the best single repository of such knowledge, although efforts are now under way within INCOSE to update them for the 21st Century. Their motivation was to provide guidance for the emerging role of system architect as the person or team responsible for coordinating engineering effort towards devising solutions to complex problems and overseeing their implementation. Rechtin and Maier observed that it was in many cases better to apply "rules of thumb" than to attempt detailed analysis, especially when this was precluded by the number of variables involved, the complexity of the interactions between stakeholders, and the internal dynamics of system solutions and the organizations responsible for their realization.

An argument in favor of the wider use of heuristics was also made by Mervyn King (2016), who was Governor of the Bank of England during the 2008 global financial crash. Looking back, he said "it is better to be roughly right than precisely wrong": banks which observed the old bankers’ rule of maintaining capital assets equal to 70% of their loan book survived, while those who relied on complex (and flawed) mathematical models of derivatives failed. He has become a powerful advocate for the use of heuristics alongside formal economics to allow bankers and others to deal with the uncertainties of global financial affairs in the modern interconnected world.

Further backing for the contemporary use of heuristics comes from Simon (1957), who coined the term "satisficing" for a situation in which people seek solutions, or accept choices or judgments, that are "good enough" for their purposes, regardless of whether they can be further optimized by precise analysis. He made the point that some heuristics were scientifically derived from experiment or systematic collection and analysis of real-world data, while others were just rules of thumb based on real-world observation or experience.

This idea has been further developed over a number of years by Gigerenzer and co-workers (for example, Gigerenzer and Selten (2001)), who have conducted research on heuristics which assist in making rapid decisions in areas of limited predictability, when probability theory is no longer helpful. They have developed a series of what they call "fast and frugal" rules of general applicability which have outperformed more conventional analysis in such areas as medical diagnosis and performance science (Raab and Gigerenzer 2015). The application to Systems Engineering has so far been little explored.

Practical Use

Experience of using heuristics suggests they should be memorable and can be most effective when phrased informally. The best examples are more than just literal expressions; they should resonate with the readers and suggest additional meaning, encouraging them to find out more about why and when to use them.

There is an underlying issue here: if you haven’t experienced the situation for which the heuristic applies, it may mean little to you, and the inherent value may well be lost; but if you already have relevant experience, you may find it obvious. The use of heuristics is therefore linked to how we learn. In a world where experience is vital, but the number of opportunities to learn on the job is limited by the number of big projects one systems engineer works on in a lifetime, the learning process has to be accelerated. Heuristics can help here if linked to other knowledge sources and accessible at the right points in a career, as part of life-long learning.

A repository of heuristics can also act as a knowledge base in its own right, especially if other media, such as video clips or training materials, or even interactive media are added to encourage discussion and feedback. Such a repository might also link to other established knowledge sources or company websites. It can be organized to reflect accepted areas of practice or in a data base tagged with metadata to allow flexible retrieval. Maier and Rechtin (2009) suggested that a repository might also act as a "reading room," allowing users to move freely among associated subjects as if they were following their curiosity in a library or a bookshop. Such a repository could also allow users to assemble a set of heuristics most meaningful to them, relevant to their personal interest or professional sphere of activity.

A further possible use of heuristics was demonstrated by Beasley, et al (2014), who used a selection in a survey to uncover how key aspects of Systems Engineering were addressed within their organization. They asked their staff to mark the heuristics according to their importance to the business and whether they were observed in practice. Analysis of the answers allowed attention to be given to areas requiring improvement.

Broadly speaking, interest in heuristics now centers on their use in two main contexts: first, encapsulating engineering knowledge in an accessible form, where the practice is widely accepted and the underlying science understood; and second, overcoming the limitations of more analytical approaches, where the science is still of limited use. In either case, a good set of heuristics requires active maintenance to reflect the evolution of practice and our constantly developing understanding of what works best. A well-curated collection of heuristics allows practitioners to retain and represent the accumulated practical wisdom of the community of the Systems Engineering profession.

This article finishes with a few simple examples applicable to early stages, with some commentary to suggest their hidden meanings and where they might lead.

• Don’t assume that the original statement of the problem is necessarily the best, or even the right one. The same point is repeated in many heuristics, such as: "The hidden assumptions are likely to be the most damaging," and "The customer may know what he wants, but not what he needs." All this has to be handled with tact and respect for the user, but experience shows that failure to reach mutual understanding early on is a fundamental cause of failure, and strong relationships forged in the course of doing such work can pay off when solving more difficult issues which might arise later on.
• In the early stages of a project, unknowns are a bigger issue than known problems. Sometimes what is being asked for is obscure, and the whole context of the systems engineering remit difficult to know, especially in areas of high uncertainty. The starting question may be less ‘What are the Requirements for this?" and more "What’s going on here?". (Kay and King 2020) Unpeeling the layers of meaning behind the second question may require methods drawn from systems thinking – looking for root causes, for example – and can take a project into unexpected areas. Again, soft skills such as empathy and intuition may be more important than those of conventional engineering.
• Model before build, wherever possible. This heuristic draws one in to the deeper question of the general use and limitation of models as a way that systems engineering tries to predict desirable and undesirable emergent system properties before committing to build it. A related heuristic states "When you test a model of a system in the real world, you validate the model not the system," and a Theorem from System Science states "The only complete model of the system is the system itself." One heuristic leads to another, leading to reflection on how systems are built in the highly uncertain world of hyperconnected IT systems, where one might postulate that "The only complete model of the system in its environment is the system in its environment," which leads into using evolutionary lifecycles, rapid deployment of prototypes, agile life cycles, and so on. The original heuristic opens a door into 21st Century systems.
• Most of the serious mistakes are made early on. This heuristic summarizes much of what has just been said. Just to show that much what is believed now might have been known for some time, here is a quote from Plato: "The beginning is the most important part of the work." (Plato 375 BCE).

Finally a heuristic about heuristics – where they come from and what they are for – is taken from a fortune cookie and provides a summary: "The work will tell you how to do it." (Wilczek 2015) The serious point being made here – apart from showing that folk wisdom can have a part to play – is that the most important lessons about how to do Systems Engineering ultimately derive from the collective experience of the community of systems engineers as they undertake their profession. Even the science used to support heuristics must prove itself in practical situations.

Works Cited

Beasley, R., A. Nolan, and A.C. Pickard. 2014. "When ‘Yes’ is the Wrong Answer." INCOSE International Symposium, Las Vegas, NV, Jun 30-Jul 3, 2014.

Churchman, C.W. 1967. "Wicked Problems". Management Science . 14(4): B-141–B-146.

Gigerenzer, G. and R. Selten (eds.). 2001. Bounded Rationality. MIT Press.

Kay, J. and M. King. 2020. Radical Uncertainty: Decision-Making for an Unknowable Future. The Bridge St. Press.

King, M. 2016. The End of Alchemy: Money, Banking and the Future of the Global Economy. New York and London: W.W. Norton and Company.

Koen, B.V. 1985. Definition of the Engineering Method. Washington, DC: American Society for Engineering Education. ISBN-0-57823-101-3.

Maier, M. and E. Rechtin. 2009. The Art of Systems Architecting, 3rd edition. CRC Press.

Plato. 375 BCE. The Republic.

Raab, M. and G. Gigerenzer. 2015. "The Power of Simplicity: A Fast-and-Frugal Heuristics Approach to Performance Science", Frontiers in Psychology, October 29, 2015.

Simon, H.A. 1957. Models of Man, Social and Rational: Mathematical Essays on Rational Human Behavior in a Social Setting. New York, NY: John Wiley and Sons.

Wilczek, F. 2015. A Beautiful Question: Finding Nature's Deep Design. New York, NY: Penguin Press.

Primary References

Gigerenzer, G. and R. Selten (eds.). 2001. Bounded Rationality . MIT Press.

Maier, M. and E. Rechtin. 2009. The Art of Systems Architecting , 3rd edition. CRC Press.

Rechtin, E. 1991. Systems Architecting: Creating and Building Complex Systems . Englewood Cliffs, NJ: Prentice-Hall.

Simon, H.A. 1957. Models of Man, Social and Rational: Mathematical Essays on Rational Human Behavior in a Social Setting . New York, NY: John Wiley and Sons.

Additional References

Royal Academy of Engineering. 2010-2011. Philosophy of Engineering, Volumes 1 and 2. Proceedings of seminars held at Royal Academy of Engineering, June 2010 and October 2011. Accessed April 9, 2021. Available: https://www.raeng.org.uk/policy/supporting-the-profession/engineering-ethics-and-philosophy/philosophy

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Using a problem-solving heuristic to teach engineering graphics

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Students often experience difficulties in developing an adequate understanding of how to solve engineering graphics problems using traditional teaching methods. Application of an explicit problem-solving technique to graphics problems can help students to understand the solution strategy. This method reinforces the details of the process, enabling students to apply the same techniques to more complicated problems. The problem-solving heuristic involves devising and evaluating a solution plan before it is implemented. Without such a solution plan, students are more likely to rush into an ill-conceived solution design without any meaningful preliminary thought. By considering a detailed solution plan for even simple problems, students should gain an in-depth understanding of the class material. This paper presents and discusses the implementation of a problem-solving approach to engineering graphics, which can be applied to both drafting and computer-aided design (CAD) exercises. Preliminary results indicate that students' skills at solving engineering graphics problems improve as a result of implementing a structured approach towards developing a solution plan.

All Science Journal Classification (ASJC) codes

• Mechanical Engineering
• Computer-aided design
• Engineering graphics
• Problem-solving heuristic

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• 10.7227/IJMEE.32.2.5

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• heuristics Social Sciences 100%
• Students Engineering & Materials Science 79%
• engineering Social Sciences 64%
• student Social Sciences 26%
• CAD Social Sciences 22%
• teaching method Social Sciences 15%
• Teaching Engineering & Materials Science 15%
• Computer aided design Engineering & Materials Science 14%

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AB - Students often experience difficulties in developing an adequate understanding of how to solve engineering graphics problems using traditional teaching methods. Application of an explicit problem-solving technique to graphics problems can help students to understand the solution strategy. This method reinforces the details of the process, enabling students to apply the same techniques to more complicated problems. The problem-solving heuristic involves devising and evaluating a solution plan before it is implemented. Without such a solution plan, students are more likely to rush into an ill-conceived solution design without any meaningful preliminary thought. By considering a detailed solution plan for even simple problems, students should gain an in-depth understanding of the class material. This paper presents and discusses the implementation of a problem-solving approach to engineering graphics, which can be applied to both drafting and computer-aided design (CAD) exercises. Preliminary results indicate that students' skills at solving engineering graphics problems improve as a result of implementing a structured approach towards developing a solution plan.

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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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Heuristics Unleashed: A Comprehensive Guide to Heuristics in Computer Science and Programming

Heuristics are nothing new, they play an important role in our daily lives, in both problem-solving and data-driven decision-making (DDDM) . As nowadays, the world is full of information, and our brains are only capable of processing a certain amount of it, heuristics help us a lot in this sense. Because if you would try to analyze every single aspect of every situation or decision, you would never get anything done.

We take thousands of decisions every single day and most of them we don’t really think about, we “know” how to behave in certain situations based on our experience and that’s what heuristics are about. When we are trying to solve a problem or make a decision, we often turn to these mental shortcuts when we need a quick solution.

Introduction to Heuristics

In the world of computer science, heuristics play a vital role in tackling complex problems and providing efficient solutions. This introductory section will explore the concept of heuristics and highlight their significance in problem-solving.

Definition of Heuristics

Heuristics is practical problem-solving techniques or methods used to find quick and effective solutions when dealing with complex issues. They are only sometimes guaranteed to produce the best or optimal solution, but they often lead to satisfactory results within a reasonable time frame. Heuristics are particularly useful when dealing with problems that have incomplete, uncertain, or imprecise information or when the search space is too ample for a complete, exhaustive search.

Importance of Heuristics in Problem Solving

The importance of heuristics in problem-solving is evident in their versatility and adaptability across different computer science domains. Some of the reasons why heuristics are crucial for problem-solving include:

• Efficiency : Heuristics can significantly reduce the time and resources needed to solve problems, especially when the search space is enormous, or the problem is too complex.
• Simplicity : Heuristic methods are often easier to implement and understand than complicated algorithms, making them more accessible to a broader audience.
• Flexibility : Heuristics can be applied to various problem domains, often with minor modifications, making them versatile problem-solving tools.
• Robustness : In situations where the available data is incomplete, uncertain, or noisy, heuristics can still provide helpful solutions, making them applicable to real-world problems.

History of Heuristics in Computer Science

Heuristics have been crucial to the growth and advancement of computer science. Here's how the history of heuristics and how they have changed over time.

Early Beginnings

Ancient Greek mathematicians and philosophers used heuristic techniques to solve mathematical and logical issues, giving rise to heuristic usage. 

Yet, with the development of artificial intelligence (AI) research in the 1950s and 1960s, heuristics were formally introduced in computer science for the first time.

Alan Turing, one of the early AI pioneers, first suggested a heuristic search in his landmark article " Computing Machines and Intelligence ." The exploration of heuristics as a tool to solve challenging problems in AI and computer science was made possible thanks to Turing's work.

George Pólya, another important person, wrote the famous book " How to Solve It " in 1945. It offered a collection of heuristics for resolving mathematical issues. Many computer scientists and researchers were motivated by Pólya's efforts to

Evolution of Heuristic Techniques

Over the years, heuristic techniques have evolved and diversified to address various complex problems in computer science. Some of the most notable developments in the history of heuristics include:

• Game Theory : In the 1950s and 1960s, researchers like Claude Shannon and Arthur Samuel developed early heuristics to explore optimal game strategies like chess and checkers. Their work paved the way for more advanced heuristic techniques used in game theory today.
• Optimization : In the 1970s, researchers began developing metaheuristic optimization algorithms, such as genetic and simulated annealing, to find near-optimal solutions to complex optimization problems.
• Machine Learning : The 1980s and 1990s witnessed significant advancements in machine learning techniques, such as decision trees and neural networks, which rely on heuristic methods to learn from data and make predictions.
• Human-Computer Interaction : Heuristic evaluation, a method for identifying usability issues in user interfaces, was introduced by Jakob Nielsen in the 1990s, highlighting the application of heuristics in human-computer interaction.
• As computer science continues to evolve, so do heuristic techniques, with researchers constantly developing new and innovative ways to apply heuristics to tackle increasingly complex problems.

So, What is Heuristic Programming?

Heuristics are mental shortcuts that help us make decisions and judgments quickly without having to spend a lot of time researching and analyzing information. They usually involve focusing on one aspect of a complex problem and ignoring others. They work well under most circumstances, but they can lead to systematic deviations from logic, probability, or rational choice. Examples that employ heuristics include using a rule of thumb, an educated guess, an intuitive judgment, a guesstimate, stereotyping, profiling, or common sense.

Heuristics in Software Development

Heuristics have become indispensable to software engineering , helping developers create more efficient, maintainable, and user-friendly software. This section will explore various ways heuristics are employed in software development.

Design Patterns

Design patterns are reusable solutions to common problems that arise in software design. They are heuristics that software developers can apply when faced with specific design challenges. By leveraging these well-established patterns, developers can create more maintainable and scalable software while avoiding common pitfalls.

Some widely used design patterns include:

• Singleton : Assures a class has just one instance and offers a universal access point to that instance.
• Factory Method : Establishes a framework for instantiating objects in a superclass, allowing subclasses to select the objects they want to use.
• Observer : Establishes a one-to-many relationship between objects so that all of its dependents are automatically informed and updated when one changes.

Agile Development

Agile development methodologies , such as Scrum and Kanban, have become famous for managing software projects more efficiently in recent years. These methodologies incorporate heuristics to streamline development, improve collaboration, and deliver high-quality software.

Some critical heuristics in agile development include:

• Iterative Development : Agile methodologies promote short development cycles, allowing teams to gather feedback continuously, identify issues, and adjust their plans accordingly.
• Time-boxing : Setting strict time limits for tasks helps teams maintain focus and prevents them from getting bogged down in unnecessary details.
• Daily Stand-ups : Regular short meetings enable team members to share updates, identify roadblocks, and ensure everyone is on the same page.
• Prioritization : Agile methodologies prioritize tasks based on their value to the end user, ensuring that the most critical features are delivered first.

Code Smells and Refactoring

Code smells indicate potential problems in the source code, often resulting from poor design choices or programming practices. Heuristics can be used to identify these code smells and guide developers in refactoring their code to improve its maintainability, readability, and overall quality.

Some examples of code smells, and corresponding refactoring techniques include:

• Long Method : A method that is too long and difficult to understand can be broken down into smaller, more manageable processes.
• Duplicate Code : Identical or similar code in multiple locations can be consolidated into a single reusable method or class.
• Feature Envy : When a method accesses data from another class more than its data, moving the technique to that class may be more appropriate.
• Software engineers can produce more effective, maintainable, and user-friendly software that satisfies the requirements of their customers and end users by utilizing heuristics in the development process.

Heuristics in Computer Sciences

In computer science, a heuristic is a problem-solving strategy or method that is not guaranteed to find the optimal solution, but is designed to find a satisfactory solution in a reasonable amount of time. Heuristics are often used in artificial intelligence, search algorithms, and optimization problems where it is not possible or impractical to use an algorithm that guarantees a correct solution.

Heuristics are often based on experience, intuition, or common sense and are used to guide the search for a solution in a way that is efficient and effective. They can be useful in situations where the problem is too complex or too large to be solved by an algorithm that guarantees a correct solution.

Examples of heuristics in computer science include:

• Best-first search
• Hill climbing
• Simulated annealing
• Genetic algorithms

Heuristics can be very useful in computer science, they can be faster and more efficient than other techniques, but they are also less reliable and might not provide optimal solutions.

In computer science, artificial intelligence , and mathematical optimization, 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. 

Heuristic programming employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal. It is important to highlight that Heuristics are the strategies derived from previous experiences with similar problems. These strategies rely on using readily accessible, though loosely applicable, information to control problem-solving in human beings, machines, and abstract issues.  And the objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. 

The trade-off criteria for deciding whether to use a heuristic for solving a given problem:

• Optimality: When several solutions exist for a given problem, does the heuristic guarantee that the best solution will be found? Is it actually necessary to find the best solution?
• Completeness: When several solutions exist for a given problem, can the heuristic find them all? Do we actually need all solutions? Many heuristics are only meant to find one solution.
• Accuracy and precision: Can the heuristic provide a confidence interval for the purported solution? Is the error bar on the solution unreasonably large?
• Execution time: Is this the best-known heuristic for solving this type of problem? Some heuristics converge faster than others. Some heuristics are only marginally quicker than classic methods.

And now, the main question is: why do we rely on heuristics?  Psychologists have suggested a few different theories:

• Effort reduction: According to this theory, people utilize heuristics as a type of cognitive laziness. Heuristics reduce the mental effort required to make choices and decisions.
• Attribute substitution: Other theories suggest people substitute simpler but related questions in place of more complex and difficult questions.
• Fast and frugal: Still other theories argue that heuristics are actually more accurate than they are biased. In other words, we use heuristics because they are fast and usually correct.

This is in contrast to algorithmic programming, which is based on mathematically provable procedures. But what is important to understand here is that Heuristic programming is characterized by programs that are self-learning; they get better with experience. Remember, heuristics don't guarantee the best or perfect solution, but they often lead to good solutions in a reasonable time frame, which is why they're so widely used in computer science.

Let us know about your thoughts and experience with heuristic programming, we would be happy to discuss it in the comments section below. 

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Heuristic Search Techniques in AI

One of the core methods AI systems use to navigate problem-solving is through heuristic search techniques. These techniques are essential for tasks that involve finding the best path from a starting point to a goal state, such as in navigation systems, game playing, and optimization problems. This article delves into what heuristic search is, its significance, and the various techniques employed in AI.

Table of Content

Understanding Heuristic Search

Significance of heuristic search in ai, components of heuristic search, types of heuristic search techniques.

• 1. A Search Algorithm*

2. Greedy Best-First Search

3. hill climbing, 4. simulated annealing, 5. beam search, applications of heuristic search, advantages of heuristic search techniques, limitations of heuristic search techniques.

Heuristics operates on the search space of a problem to find the best or closest-to-optimal solution via the use of systematic algorithms. In contrast to a brute-force approach, which checks all possible solutions exhaustively, a heuristic search method uses heuristic information to define a route that seems more plausible than the rest. Heuristics, in this case, refer to a set of criteria or rules of thumb that offer an estimate of a firm’s profitability. Utilizing heuristic guiding, the algorithms determine the balance between exploration and exploitation, and thus they can successfully tackle demanding issues. Therefore, they enable an efficient solution finding process.

The primary benefit of using heuristic search techniques in AI is their ability to handle large search spaces. Heuristics help to prioritize which paths are most likely to lead to a solution, significantly reducing the number of paths that must be explored. This not only speeds up the search process but also makes it feasible to solve problems that are otherwise too complex to handle with exact algorithms.

Heuristic search algorithms typically comprise several essential components:

• State Space: This implies that the totality of all possible states or settings, which is considered to be the solution for the given problem.
• Initial State: The instance in the search tree of the highest level with no null values, serving as the initial state of the problem at hand.
• Goal Test: The exploration phase ensures whether the present state is a terminal or consenting state in which the problem is solved.
• Successor Function: This create a situation where individual states supplant the current state which represent the possible moves or solutions in the problem space.
• Heuristic Function: The function of a heuristic is to estimate the value or distance from a given state to the target state. It helps to focus the process on regions or states that has prospect of achieving the goal.

Over the history of heuristic search algorithms, there have been a lot of techniques created to improve them further and attend different problem domains. Some prominent techniques include:

1. A Search Algorithm *

A* Search Algorithm is perhaps the most well-known heuristic search algorithm. It uses a best-first search and finds the least-cost path from a given initial node to a target node. It has a heuristic function, often denoted as [Tex]f(n) = g(n) + h(n)[/Tex] , where g(n) is the cost from the start node to n , and h(n) is a heuristic that estimates the cost of the cheapest path from n to the goal. A* is widely used in pathfinding and graph traversal.

Greedy best-first search expands the node that is closest to the goal, as estimated by a heuristic function. Unlike A*, which takes into account the cost of the path from the start node to the current node, the greedy best-first search only prioritizes the estimated cost from the current node to the goal. This makes it faster but less optimal than A*.

Hill climbing is a heuristic search used for mathematical optimization problems. It is a variant of the gradient ascent method. Starting from a random point, the algorithm takes steps in the direction of increasing elevation or value to find the peak of the mountain or the optimal solution to the problem. However, it may settle for a local maximum and not reach the global maximum.

Inspired by the process of annealing in metallurgy, simulated annealing is a probabilistic technique for approximating the global optimum of a given function. It allows the algorithm to jump out of any local optimums in search of the global optimum by probabilistically deciding whether to accept or reject a higher-cost solution during the early phases of the search.

Beam search is a heuristic search algorithm that explores a graph by expanding the most promising nodes in a limited set or “beam”. The beam width, which limits the number of nodes stored in memory, plays a crucial role in the performance and accuracy of the search.

Heuristic search techniques find application in a wide range of problem-solving scenarios, including:

• Pathfinding: Discovery, of the shortest distance that can be found from the start point to the destination at the point of coordinates or graph.
• Optimization: Solving the problem of the optimal distribution of resources, planning or posting to achieve maximum results.
• Game Playing: The agency of AI with some board games, e.g., chess or Go, is on giving guidance and making strategy-based decisions to the agents.
• Robotics: Scheduling robots` location and movement to guide carefully expeditions and perform given tasks with high efficiency.
• Natural Language Processing: Language processing tasks involving search algorithms, such as parsing or semantic analysis, should be outlined. That means.

Heuristic search techniques offer several advantages:

• Efficiency: As they are capable of aggressively digesting large areas for the more promising lines, they can allot more time and resources to investigate the area.
• Optimality: If the methods that an algorithm uses are admissible, A* guarantees of an optimal result.
• Versatility: Heuristic search methods encompass a spectrum of problems that are applied to various domains of problems.
• Heuristic Quality: The power of heuristic search strongly depends on the quality of function the heuristic horizon. If the heuristics are constructed thoughtlessly, then their level of performance may be low or inefficient.
• Space Complexity: The main requirement for some heuristic search algorithms could be a huge memory size in comparison with the others, especially in cases where the search space considerably increases.
• Domain-Specificity: It is often the case that devising efficient heuristics depends on the specifics of the domain, a challenging obstruction to development of generic approaches.

Heuristic search methodologies open into the AI toolbox, which enables the refined exploration and traversal of perplexing issue areas equipped with a precise analytic precision that is unquenchable in the midst of challenging circumstances. Using heuristic-based approach, these algorithms can achieve an equilibrium between exploring the solution space and exploiting what is already known and get near-optimal solution to at least some domains. With the development of AI even further towards perfection, new strides in progress with heuristic search algorithms will undoubtedly lead to the exploration of more effective methods for problems resolving and decision making.

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Why do we take mental shortcuts?

What are heuristics.

Heuristics are mental shortcuts that can facilitate problem-solving and probability judgments. These strategies are generalizations, or rules-of-thumb, that reduce cognitive load. They can be effective for making immediate judgments, however, they often result in irrational or inaccurate conclusions.

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We use heuristics in all sorts of situations. One type of heuristic, the availability heuristic , often happens when we’re attempting to judge the frequency with which a certain event occurs. Say, for example, someone asked you whether more tornadoes occur in Kansas or Nebraska. Most of us can easily call to mind an example of a tornado in Kansas: the tornado that whisked Dorothy Gale off to Oz in Frank L. Baum’s The Wizard of Oz . Although it’s fictional, this example comes to us easily. On the other hand, most people have a lot of trouble calling to mind an example of a tornado in Nebraska. This leads us to believe that tornadoes are more common in Kansas than in Nebraska. However, the states actually report similar levels. 1

Individual effects

The thing about heuristics is that they aren’t always wrong. As generalizations, there are many situations where they can yield accurate predictions or result in good decision-making. However, even if the outcome is favorable, it was not achieved through logical means. When we use heuristics, we risk ignoring important information and overvaluing what is less relevant. There’s no guarantee that using  heuristics will work out and, even if it does, we’ll be making the decision for the wrong reason. Instead of basing it on reason, our behavior is resulting from a mental shortcut with no real rationale to support it.

Systemic effects

Heuristics become more concerning when applied to politics, academia, and economics. We may all resort to heuristics from time to time, something that is true even of members of important institutions who are tasked with making large, influential decisions. It is necessary for these figures to have a comprehensive understanding of the biases and heuristics that can affect our behavior, so as to promote accuracy on their part.

How it affects product

Heuristics can be useful in product design. Specifically, because heuristics are intuitive to us, they can be applied to create a more user-friendly experience and one that is more valuable to the customer. For example, color psychology is a phenomenon explaining how our experiences with different colors and color families can prime certain emotions or behaviors. Taking advantage of the representativeness heuristic, one could choose to use passive colors (blue or green) or more active colors (red, yellow, orange) depending on the goals of the application or product. 18 For example, if a developer is trying to evoke a feeling of calm for their app that provides guided meditations, they may choose to make the primary colors of the program light blues and greens. Colors like red and orange are more emotionally energizing and may be useful in settings like gyms or crossfit programs. 

By integrating heuristics into products we can enhance the user experience. If an application, device, or item includes features that make it feel intuitive, easy to navigate and familiar, customers will be more inclined to continue to use it and recommend it to others. Appealing to those mental shortcuts we can minimize the chances of user error or frustration with a product that is overly complicated.

Heuristics and AI

Artificial intelligence and machine learning tools already use the power of heuristics to inform its output. In a nutshell, simple AI tools operate based on a set of built in rules and sometimes heuristics! These are encoded within the system thus aiding in decision-making and the presentation of learning material. Heuristic algorithms can be used to solve advanced computational problems, providing efficient and approximate solutions.  Like in humans, the use of heuristics can result in error, and thus must be used with caution. However, machine learning tools and AI can be useful in supporting human decision-making, especially when clouded by emotion, bias or irrationality due to our own susceptibility to heuristics. 

Why it happens

In their paper “Judgment Under Uncertainty: Heuristics and Biases” 2 , Daniel Kahneman and Amos Tversky identified three different kinds of heuristics: availability, representativeness, as well as anchoring and adjustment. Each type of heuristic is used for the purpose of reducing the mental effort needed to make a decision, but they occur in different contexts.

Availability heuristic

The availability heuristic, as defined by Kahneman and Tversky, is the mental shortcut used for making frequency or probability judgments based on “the ease with which instances or occurrences can be brought to mind”. 3 This was touched upon in the previous example, judging the frequency with which tornadoes occur in Kansas relative to Nebraska. 3

The availability heuristic occurs because certain memories come to mind more easily than others. In Kahneman and Tversky’s example participants were asked if more words in the English language start with the letter K or have K as the third letter  Interestingly, most participants responded with the former when in actuality, it is the latter that is true. The idea being that it is much more difficult to think of words that have K as the third letter than it is to think of words that start with K. 4 In this case,  words that begin with K are more readily available to us than words with the K as the third letter.

Representativeness heuristic

Individuals tend to classify events into categories, which, as illustrated by Kahneman and Tversky, can result in our use of the representativeness heuristic. When we use this heuristic, we categorize events or objects based on how they relate to instances we are already familiar with.  Essentially, we have built our own categories, which we use to make predictions about novel situations or people. 5 For example, if someone we meet in one of our university lectures looks and acts like what we believe to be a stereotypical medical student, we may judge the probability that they are studying medicine as highly likely, even without any hard evidence to support that assumption.

The representativeness heuristic is associated with prototype theory. 6 This prominent theory in cognitive science, the prototype theory explains object and identity recognition. It suggests that we categorize different objects and identities in our memory. For example, we may have a category for chairs, a category for fish, a category for books, and so on. Prototype theory posits that we develop prototypical examples for these categories by averaging every example of a given category we encounter. As such, our prototype of a chair should be the most average example of a chair possible, based on our experience with that object. This process aids in object identification because we compare every object we encounter against the prototypes stored in our memory. The more the object resembles the prototype, the more confident we are that it belongs in that category. 

Prototype theory may give rise to the representativeness heuristic as it is in situations when a particular object or event is viewed as similar to the prototype stored in our memory, which leads us to classify the object or event into the category represented by that prototype. To go back to the previous example, if your peer closely resembles your prototypical example of a med student, you may place them into that category based on the prototype theory of object and identity recognition. This, however, causes you to commit the representativeness heuristic.

Anchoring and adjustment heuristic

Another heuristic put forth by Kahneman and Tversky in their initial paper is the anchoring and adjustment heuristic. 7 This heuristic describes how, when estimating a certain value, we tend to give an initial value, then adjust it by increasing or decreasing our estimation. However, we often get stuck on that initial value – which is referred to as anchoring – this results in us making insufficient adjustments. Thus, the adjusted value is biased in favor of the initial value we have anchored to.

In an example of the anchoring and adjustment heuristic, Kahneman and Tversky gave participants questions such as “estimate the number of African countries in the United Nations (UN).” A wheel labeled with numbers from 0-100 was spun, and participants were asked to say whether or not the number the wheel landed on was higher or lower than their answer to the question. Then, participants were asked to estimate the number of African countries in the UN, independent from the number they had spun. Regardless, Kahneman and Tversky found that participants tended to anchor onto the random number obtained by spinning the wheel. The results showed that  when the number obtained by spinning the wheel was 10, the median estimate given by participants was 25, while, when the number obtained from the wheel was 65, participants’ median estimate was 45.8.

A 2006 study by Epley and Gilovich, “The Anchoring and Adjustment Heuristic: Why the Adjustments are Insufficient” 9 investigated the causes of this heuristic. They illustrated that anchoring often occurs because the new information that we anchor to is more accessible than other information Furthermore, they provided empirical evidence to demonstrate that our adjustments tend to be insufficient because they require significant mental effort, which we are not always motivated to dedicate to the task. They also found that providing incentives for accuracy led participants to make more sufficient adjustments. So, this particular heuristic generally occurs when there is no real incentive to provide an accurate response.

Quick and easy

Though different in their explanations, these three types of heuristics allow us to respond automatically without much effortful thought. They provide an immediate response and do not use up much of our mental energy, which allows us to dedicate mental resources to other matters that may be more pressing. In that way, heuristics are efficient, which is a big reason why we continue to use them. That being said, we should be mindful of how much we rely on them because there is no guarantee of their accuracy.

Why it is important

As illustrated by Tversky and Kahneman, using heuristics can cause us to engage in various cognitive biases and commit certain fallacies. 10 As a result, we may make poor decisions, as well as inaccurate judgments and predictions. Awareness of heuristics can aid us in avoiding them, which will ultimately lead us to engage in more adaptive behaviors.

How to avoid it

Heuristics arise from automatic System 1 thinking. It is a common misconception that errors in judgment can be avoided by relying exclusively on System 2 thinking. However, as pointed out by Kahneman, neither System 2 nor System 1 are infallible. 11   While System 1 can result in relying on heuristics leading to certain biases, System 2 can give rise to other biases, such as the confirmation bias . 12 In truth, Systems 1 and 2 complement each other, and using them together can lead to more rational decision-making. That is, we shouldn’t make judgments automatically, without a second thought, but we shouldn’t overthink things to the point where we’re looking for specific evidence to support our stance. Thus, heuristics can be avoided by making judgments more effortfully, but in doing so, we should attempt not to overanalyze the situation.

How it all started

The first three heuristics – availability, representativeness, as well as anchoring and adjustment – were identified by Tverksy and Kahneman in their 1974 paper, “Judgment Under Uncertainty: Heuristics and Biases”. 13 In addition to presenting these heuristics and their relevant experiments, they listed the respective biases each can lead to.

For instance, upon defining the availability heuristic, they demonstrated how it may lead to illusory correlation , which is the erroneous belief that two events frequently co-occur. Kahneman and Tversky made the connection by illustrating how the availability heuristic can cause us to over- or under-estimate the frequency with which certain events occur. This may result in drawing correlations between variables when in reality there are none.  

Referring to our tendency to overestimate our accuracy making probability judgments, Kahneman and Tversky also discussed how the illusion of validity is facilitated by the representativeness heuristic. The more representative an object or event is, the more confident we feel in predicting certain outcomes. The illusion of validity, as it works with the representativeness heuristic, can be demonstrated by our assumptions of others based on past experiences. If you have only ever had good experiences with people from Canada, you will be inclined to judge most Canadians as pleasant. In reality, your small sample size cannot account for the whole population. Representativeness is not the only factor in determining the probability of an outcome or event, meaning we should not be as confident in our predictive abilities.

Example 1 – Advertising

Those in the field of advertising should have a working understanding of heuristics as consumers often rely on these shortcuts when making decisions about purchases. One heuristic that frequently comes into play in the realm of advertising is the scarcity heuristic . When assessing the value of something, we often fall back on this heuristic, leading us to believe that the rarity or exclusiveness of an object contributes to its value.

A 2011 study by Praveen Aggarwal, Sung Yul Jun, and Jong Ho Huh evaluated the impact of “scarcity messages” on consumer behavior. They found that both “limited quantity” and “limited time” advertisements influence consumers’ intentions to purchase, but “limited quantity” messages are more effective. This explains why people get so excited over the one-day-only Black Friday sales, and why the countdowns of units available on home shopping television frequently lead to impulse buys. 14

Knowledge of the scarcity heuristic can help businesses thrive, as “limited quantity” messages make potential consumers competitive and increase their intentions to purchase. 15 This marketing technique can be a useful tool for bolstering sales and bringing attention to your business.

Example 2 – Stereotypes

One of the downfalls of heuristics is that they have the potential to lead to stereotyping, which is often harmful. Kahneman and Tversky illustrated how the representativeness heuristic might result in the propagation of stereotypes. The researchers presented participants with a personality sketch of a fictional man named Steve followed by a list of possible occupations. Participants were tasked with ranking the likelihood of each occupation being Steve’s. Since the personality sketch described Steve as shy, helpful, introverted, and organized, participants tended to indicate that it was probable that he was a  librarian. 16 In this particular case the stereotype is less harmful than many others, however it accurately illustrates the link between heuristics and stereotypes.

Published in 1989, Patricia Devine’s paper “Stereotypes and Prejudice: Their Automatic and Controlled Components” illustrates how, even among people who are low in prejudice, rejecting stereotypes requires a certain level of motivation and cognitive capacity. 17 We typically use heuristics in order to avoid exerting too much mental energy, specifically when we are not sufficiently motivated to dedicate mental resources to the task at hand. Thus, when we lack the mental capacity to make a judgment or decision effortfully, we may rely upon automatic heuristic responses and, in doing so, risk propagating stereotypes.

Stereotypes are an example of how heuristics can go wrong. Broad generalizations do not always apply, and their continued use can have serious consequences. This underscores the importance of effortful judgment and decision-making, as opposed to automatic.

Heuristics are mental shortcuts that allow us to make quick judgment calls based on generalizations or rules of thumb.

Heuristics, in general, occur because they are efficient ways of responding when we are faced with problems or decisions. They come about automatically, allowing us to allocate our mental energy elsewhere. Specific heuristics occur in different contexts; the availability heuristic happens because we remember certain memories better than others, the representativeness heuristic can be explained by prototype theory, and the anchoring and adjustment heuristic occurs due to lack of incentive to put in the effort required for sufficient adjustment.

The scarcity heuristic, which refers to how we value items more when they are limited, can be used to the advantage of businesses looking to increase sales. Research has shown that advertising objects as “limited quantity” increases consumers' competitiveness and their intentions to buy the item.

While heuristics can be useful, we should exert caution, as they are generalizations that may lead us to propagate stereotypes ranging from inaccurate to harmful.

Putting more effort into decision-making instead of making decisions automatically can help us avoid heuristics. Doing so requires more mental resources, but it will lead to more rational choices.

Related TDL articles

What are heuristics.

This interview with The Decision Lab’s Managing Director Sekoul Krastev delves into the history of heuristics, their applications in the real world, and their consequences, both positive and negative.

10 Decision-Making Errors that Hold Us Back at Work

In this article, Dr. Melina Moleskis examines the common decision-making errors that occur in the workplace. Everything from taking in feedback provided by customers to cracking the problems of on-the-fly decision-making, Dr. Moleskis delivers workable solutions that anyone can implement. 

• Gilovich, T., Keltner, D., Chen. S, and Nisbett, R. (2015).  Social Psychology  (4th edition). W.W. Norton and Co. Inc.
• Tversky, A. and Kahneman, D. (1974). Judgment Under Uncertainty: Heuristics and Biases.  Science . 185(4157), 1124-1131.
• Mervis, C. B., & Rosch, E. (1981). Categorization of natural objects.  Annual Review of Psychology ,  32 (1), 89–115. https://doi.org/10.1146/annurev.ps.32.020181.000513
• Epley, N., & Gilovich, T. (2006). The anchoring-and-adjustment heuristic.  Psychological Science -Cambridge- ,  17 (4), 311–318.
• System 1 and System 2 Thinking.  The Marketing Society.  https://www.marketingsociety.com/think-piece/system-1-and-system-2-thinking
• Aggarwal, P., Jun, S. Y., & Huh, J. H. (2011). Scarcity messages.  Journal of Advertising ,  40 (3), 19–30.
• Devine, P. G. (1989). Stereotypes and prejudice: their automatic and controlled components.  Journal of Personality and Social Psychology ,  56 (1), 5–18. https://doi.org/10.1037/0022-3514.56.1.5
• Kuo, L., Chang, T., & Lai, C.-C. (2022). Research on product design modeling image and color psychological test. Displays, 71, 102108. https://doi.org/10.1016/j.displa.2021.102108

About the Authors

Dan is a Co-Founder and Managing Director at The Decision Lab. He is a bestselling author of Intention - a book he wrote with Wiley on the mindful application of behavioral science in organizations. Dan has a background in organizational decision making, with a BComm in Decision & Information Systems from McGill University. He has worked on enterprise-level behavioral architecture at TD Securities and BMO Capital Markets, where he advised management on the implementation of systems processing billions of dollars per week. Driven by an appetite for the latest in technology, Dan created a course on business intelligence and lectured at McGill University, and has applied behavioral science to topics such as augmented and virtual reality.

Dr. Sekoul Krastev

Sekoul is a Co-Founder and Managing Director at The Decision Lab. He is a bestselling author of Intention - a book he wrote with Wiley on the mindful application of behavioral science in organizations. A decision scientist with a PhD in Decision Neuroscience from McGill University, Sekoul's work has been featured in peer-reviewed journals and has been presented at conferences around the world. Sekoul previously advised management on innovation and engagement strategy at The Boston Consulting Group as well as on online media strategy at Google. He has a deep interest in the applications of behavioral science to new technology and has published on these topics in places such as the Huffington Post and Strategy & Business.

Hindsight Bias

Why do unpredictable events only seem predictable after they occur, hot hand fallacy, why do we expect previous success to lead to future success, hyperbolic discounting, why do we value immediate rewards more than long-term rewards.

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An efficient ODE-solving method based on heuristic and statistical computations: αII-(2 + 3)P method

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The main goal of this paper is to develop a new class of sixth- and fourth-order implicit Runge–Kutta methods to treat linear and nonlinear initial value ordinary differential equations (ODEs) in applied science and engineering. In this regard, a completely unconventional approach is taken to derive the new formulations. They are derived through a computational approach without involving in the rigorous theory of order conditions. The new formulation has two distinctive parts, namely integration and interpolation. In the first part, we introduce a form of multi-stage single-step numerical integrator with unknown weights. These weights are determined by four simple interconnectivity rules that relate them to minimize the error of the integrator. Some of these rules are based on our expertise gathered from the available prominent quadratures. However, the most important one, the so-called fundamental weighting rule (FWR), is obtained from evolutionary and statistical computations. The second part of this research focuses on developing efficient tools for evaluating the internal stages of the integrator. Accordingly, a series of strong Hermite interpolators has been extended to approximate the solution values at the internal stages. They effectively collaborate with the integrator in several numerical algorithms, the so-called \(\alpha II-(q+r)P\) algorithms, to solve applied ODEs. Development of the integrator, its weighting rules, and the interpolators constitutes the central contribution of this research. Numerical studies demonstrate that the presented algorithms exhibit a high level of accuracy when approximating high-frequency problems as well as long-term solutions of linear and nonlinear ODEs.

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Acknowledgements

I would like to express my sincere thanks to the reviewers of this paper for their valuable feedback and constructive comments. Their insightful suggestions greatly improved the scientific quality of this work. I am truly appreciative of their time and expertise in reviewing this manuscript.

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Babaei, M. An efficient ODE-solving method based on heuristic and statistical computations: αII-(2 + 3)P method. J Supercomput (2024). https://doi.org/10.1007/s11227-024-06137-2

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Accepted : 09 April 2024

Published : 31 May 2024

DOI : https://doi.org/10.1007/s11227-024-06137-2

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