problem solving and visualisation

Mindfulness

How visualization can benefit your well-being, visualization can help you reach a range of goals..

Updated November 20, 2023 | Reviewed by Gary Drevitch

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Co-written by Kelsey Schultz and Tchiki Davis

Visualization, also called mental imagery , is essentially seeing with the mind’s eye or hearing with the mind’s ear. That is, when visualizing you are having a visual sensory experience without the use of your eyes. In fact, research has shown that visualization recruits the same brain areas that actual seeing does (Pearson et al., 2015).

Humans have evolved to rely heavily on our eyesight, making us highly visually-oriented creatures. Because our brains are adapted to easily process and comprehend visual information, visualization can be a powerful tool for influencing our thoughts, emotions , and behaviors. In fact, research has shown that processing emotions using visualization is more powerful than processing verbally (Blackwell et al., 2019). For example, when research participants listen to descriptions of emotionally valenced situations (i.e., “your boss telling you that they are disappointed with your work”), participants who are instructed to imagine themselves in the situation demonstrate a greater change in mood than those that are instructed only to think about the situation verbally (Blackwell et al., 2019).

There appear to be a number of emotional, cognitive, and behavioral benefits to practicing visualization.

​Emotional. Some forms of visualization have been shown to increase optimism and other positive emotions (Murphy et al., 2015). It has also been shown to be a useful method for regulating negative emotions such as anxiety or overwhelm (Blackwell et al., 2019).

Cognitive. Visualization techniques can be used to facilitate some kinds of decision-making and problem-solving (Blackwell et al., 2019). For example, visualization might be helpful when planning the best route to take on your upcoming road trip. Visualization techniques, such as the mind palace, are also an effective means of improving memory . The mind palace technique involves using a place you are very familiar with, such as your bedroom, and using different locations within that space as mnemonic devices associated with a particular piece of information you are trying to store.

Behavioral. Visualization can also help us achieve our goals by allowing us to determine the appropriate sequences of actions needed to reach our goal and identify any potential obstacles we might encounter as we proceed toward a goal. In other words, we can use visualization as a sort of rough draft for our plans by imagining each step we need to take to reach our goal, what each step might include, what might go wrong, and the ways in which we might need to prepare.

Visualization Tools

Music. Visualization music is music that is specifically intended to facilitate visualization and similar meditative processes. This kind of music can also be described as atmospheric or ambient, as the purpose is not to occupy your attention , but rather to help you focus attention on your visualizations.

Boards. Visualization boards, also called vision boards , are visual representations of your goals , intentions, and desires. Vision boards are typically poster-sized and include a collage-type arrangement of images that symbolize different facets of your goals and intentions. Vision boards are useful for ensuring that your goals remain salient. That is, by creating a visual representation of your goals, you can easily look back at your vision board and remind yourself of the intentions you set. When your intentions are at the forefront of your mind, you are more likely to act in accordance with them.

Guided Imagery

Guided imagery is a visualization exercise in which you engage all of your senses as you imagine yourself in a positive, peaceful environment.

• To begin, find a comfortable position, close your eyes, and begin breathing slowly and deeply as you start to relax.
• Next, visualize a place where you feel calm and content. This can be a place you’ve been before, a place you would like to go, or a place that is wholly the product of your imagination . Engage all of your senses to add depth and detail to the place you are visualizing. Can you feel a soft breeze? Do you hear birds or the sound of water lapping on the shore?
• Reflect on the calm that emerges as you move deeper into your vision.
• As you inhale, imagine peace washing over you and filling your body.
• As you exhale, imagine exhaustion, tension , and stress being washed away.
• Stay in your vision for as long as you like.

Visualization is a simple yet powerful technique that we can use to improve many facets of our lives. We can use visualization to improve our mood, help us remember important information, facilitate problem-solving and decision-making , and boost progress toward our goals. Depending on the purpose, there are many forms of visualization we can practice. For example, if we are trying to regulate our mood we might try visualization meditation , whereas if we are trying to solidify our goals for the new year we might use a vision board or a mind map .

Adapted from a post on visualization published by The Berkeley Well-Being Institute.

Blackwell, S. E. (2019). Mental imagery: From basic research to clinical practice. Journal of Psychotherapy Integration, 29(3), 235.

Murphy, S. E., O’Donoghue, M. C., Drazich, E. H., Blackwell, S. E., Nobre, A. C., & Holmes, E. A. (2015). Imagining a brighter future: the effect of positive imagery training on mood, prospective mental imagery and emotional bias in older adults. Psychiatry Research, 230(1), 36-43.

Pearson, J., Naselaris, T., Holmes, E. A., & Kosslyn, S. M. (2015). Mental imagery: functional mechanisms and clinical applications. Trends in cognitive sciences, 19(10), 590-602.​

Tchiki Davis, Ph.D. , is a consultant, writer, and expert on well-being technology.

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Thinking Through, and By, Visualising

This article first appeared in Mathematics Teaching 207, published by the Association of Teachers of Mathematics .

We rely on visualising when we solve problems. Sometimes we create an image of the situation that is being discussed in order to make sense of it; sometimes we need to visualise a model that can represent the situation mathematically before we can begin to develop it, and sometimes we visualise to see 'what will happen if ...?'. But are there other ways in which we visualise when solving mathematical problems and if so how can we encourage, value and develop visualising in our classrooms?

This article is based on some of the ideas that emerged during the production of a book (and accompanying CD) which takes visualising as its focus (Piggott and Pumfrey, 2007). It was while we were working on this book we began to identify problems which helped us to take a structured view of the purposes and skills of visualising that we thought worth sharing with you.

Some background

We often associate visualising in mathematics with drawing pictures or diagrams as an aid to getting started on problems. But visualising has a much wider role to play in problem solving including supporting the development of ideas and facilitating communication of results and understanding. In these senses it is not just about pictures and diagrams. For example, sometimes we place particular emphasis on using visualising to help understand and develop a plan to solve a problem. In producing such a visualisation, the problem solver is identifying the key components of the problem and the relationships between them. This process has two main elements:

• an internal model or visualisation (described as imagery by Crapo et al. (2000))
• an external representation (described as a visualisation by Crapo et al. (2000)).

Crapo et al. emphasise the importance of the interplay between these internal and external representations which support the development of an effective model. Key to this is both the physical representation which sparks ideas about how the model can be improved, and also discussion and communication with others to come to a shared visualisation. If we want learners to utilise and improve their capacity to visualise, we need to identify why visualisation is important (the purposes of visualising) and what visualising skills we want our pupils to develop. To do this we need to know what a visualising opportunity is, make that opportunity available and have a language with which to talk about it.

Purposes of visualising

We have been able to identify three purposes for visualising:

• to step into a problem,
• to plan ahead.

Visualising to step into the problem:

Here visualisations are used to help with understanding what the problem is about. The visualisation gives pupils the space to go deep into the situation to clarify and support their understanding before any generalisation can happen. For example:

The pupils are led through the context with a simple example. The teacher models the process being investigated with frequent pauses to check for understanding. Whilst the teacher is demonstrating, the pupils are asked to visualise and describe what will happen next. The teacher's actions give a focus, and a motivation for the visualisation and enable immediate feedback. The use of language to explain what is seen 'in the mind's eye' also helps with reinforcing the process. Pupils are being encouraged to make sense of the situation through the visualisation.

Visualising to model a situation:

This is particularly useful when the situation is physically unattainable, in other words to try to see the 'unseeable', for example the inside of a 3D object, or considering a case involving a very large number. To illustrate this, have a look at the problem Cubes Within Cubes .

In this example it is not possible to 'see inside' a large cube and, although smaller cubes can be constructed as the layers increase, it is not possible to see the centre cube at the same time as the surrounding layer. The modelling in this problem is two-fold. Initially the pupils are encouraged to think in stages and later to use the visualising strategies to tackle the main problem. An outcome of the lesson can be pupil production of story boards as representations of the visualisations and stages they used whilst problem solving. These images have two aspects:

• they reflect the imagery the pupils used,
• they represent the route the pupils may have taken to obtain their solution.

Visualising to plan ahead:

Visualising skills.

• Being able to compare other people's representations with our own. For example in Cubes Within Cubes , comparing the two visualisations given above with your own.
• Being able to identify the general and the specific in a representation and their significance in terms of the problem at hand. For example in a problem involving polyominoes, you might start with a domino (from which you are going to build triominoes), asking learners to visualise putting two squares together. Recognition that there is an infinite number of orientations (general) is important but, in preparing to build triominoes, you only need to consider one orientation and to have that very clearly in your mind.
• Trying to hold more than one image in your head. For example: remembering a starting point and being able to 'rewind' when a sequence of moves does not work; keeping more than one aspect of the problem in 'view'. For example in the problem Roundabout the visualisation requires you to be thinking about both the circle moving round and the locus its centre makes.

Problem-Solving Flowchart: A Visual Method to Find Perfect Solutions

Lucid Content

Reading time: about 7 min

“People ask me questions Lost in confusion Well, I tell them there's no problem Only solutions” —John Lennon, “Watching the Wheels”

Despite John Lennon’s lyrics, nobody is free from problems, and that’s especially true in business. Chances are that you encounter some kind of problem at work nearly every day, and maybe you’ve had to “put out a fire” before lunchtime once or twice in your career.

But perhaps what Lennon’s saying is that, no matter what comes our way, we can find solutions. How do you approach problems? Do you have a process in place to ensure that you and your co-workers come to the right solution?

In this article, we will give you some tips on how to find solutions visually through a problem-solving flowchart and other methods.

What is visual problem-solving?

If you are a literal thinker, you may think that visual problem-solving is something that your ophthalmologist does when your vision is blurry. For the rest of us, visual problem-solving involves executing the following steps in a visual way:

• Define the problem.
• Brainstorm solutions.
• Pick a solution.
• Implement solutions.
• Review the results.

How to make your problem-solving process more visual

Words pack a lot of power and are very important to how we communicate on a daily basis. Using words alone, you can brainstorm, organize data, identify problems, and come up with possible solutions. The way you write your ideas may make sense to you, but it may not be as easy for other team members to follow.

When you use flowcharts, diagrams, mind maps, and other visuals, the information is easier to digest. Your eyes dart around the page quickly gathering information, more fully engaging your brain to find patterns and make sense of the data.

Identify the problem with mind maps

So you know there is a problem that needs to be solved. Do you know what that problem is? Is there only one problem? Is the problem sum total of a bunch of smaller problems?

You need to ask these kinds of questions to be sure that you are working on the root of the issue. You don’t want to spend too much time and energy solving the wrong problem.

To help you identify the problem, use a mind map. Mind maps can help you visually brainstorm and collect ideas without a strict organization or structure. A mind map more closely aligns with the way a lot of our brains work—participants can bounce from one thought to the next defining the relationships as they go.

Mind mapping to solve a problem includes, but is not limited to, these relatively easy steps:

• In the center of the page, add your main idea or concept (in this case, the problem).
• Branch out from the center with possible root causes of the issue. Connect each cause to the central idea.
• Branch out from each of the subtopics with examples or additional details about the possible cause. As you add more information, make sure you are keeping the most important ideas closer to the main idea in the center.
• Use different colors, diagrams, and shapes to organize the different levels of thought.

Alternatively, you could use mind maps to brainstorm solutions once you discover the root cause. Search through Lucidchart’s mind maps template library or add the mind map shape library to quickly start your own mind map.

Create a problem-solving flowchart

A mind map is generally a good tool for non-linear thinkers. However, if you are a linear thinker—a person who thinks in terms of step-by-step progression making a flowchart may work better for your problem-solving strategy. A flowchart is a graphical representation of a workflow or process with various shapes connected by arrows representing each step.

Whether you are trying to solve a simple or complex problem, the steps you take to solve that problem with a flowchart are easy and straightforward. Using boxes and other shapes to represent steps, you connect the shapes with arrows that will take you down different paths until you find the logical solution at the end.

Flowcharts or decision trees are best used to solve problems or answer questions that are likely to come up multiple times. For example, Yoder Lumber , a family-owned hardwood manufacturer, built decision trees in Lucidchart to demonstrate what employees should do in the case of an injury.

To start your problem-solving flowchart, follow these steps:

• Draw a starting shape to state your problem.
• Draw a decision shape where you can ask questions that will give you yes-or-no answers.
• Based on the yes-or-no answers, draw arrows connecting the possible paths you can take to work through the steps and individual processes.
• Continue following paths and asking questions until you reach a logical solution to the stated problem.
• Try the solution. If it works, you’re done. If it doesn’t work, review the flowchart to analyze what may have gone wrong and rework the flowchart until you find the solution that works.

If your problem involves a process or workflow , you can also use flowcharts to visualize the current state of your process to find the bottleneck or problem that’s costing your company time and money.

Lucidchart has a large library of flowchart templates to help you analyze, design, and document problem-solving processes or any other type of procedure you can think of.

Draw a cause-and-effect diagram

A cause-and-effect diagram is used to analyze the relationship between an event or problem and the reason it happened. There is not always just one underlying cause of a problem, so this visual method can help you think through different potential causes and pinpoint the actual cause of a stated problem.

Cause-and-effect diagrams, created by Kaoru Ishikawa, are also known as Ishikawa diagrams, fishbone diagrams , or herringbone diagrams (because they resemble a fishbone when completed). By organizing causes and effects into smaller categories, these diagrams can be used to examine why things went wrong or might go wrong.

To perform a cause-and-effect analysis, follow these steps.

1. Start with a problem statement.

The problem statement is usually placed in a box or another shape at the far right of your page. Draw a horizontal line, called a “spine” or “backbone,” along the center of the page pointing to your problem statement.

2. Add the categories that represent possible causes.

For example, the category “Materials” may contain causes such as “poor quality,” “too expensive,” and “low inventory.” Draw angled lines (or “bones”) that branch out from the spine to these categories.

3. Add causes to each category.

Draw as many branches as you need to brainstorm the causes that belong in each category.

Like all visuals and diagrams, a cause-and-effect diagram can be as simple or as complex as you need it to be to help you analyze operations and other factors to identify causes related to undesired effects.

Collaborate with Lucidchart

You may have superior problem-solving skills, but that does not mean that you have to solve problems alone. The visual strategies above can help you engage the rest of your team. The more involved the team is in the creation of your visual problem-solving narrative, the more willing they will be to take ownership of the process and the more invested they will be in its outcome.

In Lucidchart, you can simply share the documents with the team members you want to be involved in the problem-solving process. It doesn’t matter where these people are located because Lucidchart documents can be accessed at any time from anywhere in the world.

Whatever method you decide to use to solve problems, work with Lucidchart to create the documents you need. Sign up for a free account today and start diagramming in minutes.

Lucidchart, a cloud-based intelligent diagramming application, is a core component of Lucid Software's Visual Collaboration Suite. This intuitive, cloud-based solution empowers teams to collaborate in real-time to build flowcharts, mockups, UML diagrams, customer journey maps, and more. Lucidchart propels teams forward to build the future faster. Lucid is proud to serve top businesses around the world, including customers such as Google, GE, and NBC Universal, and 99% of the Fortune 500. Lucid partners with industry leaders, including Google, Atlassian, and Microsoft. Since its founding, Lucid has received numerous awards for its products, business, and workplace culture. For more information, visit lucidchart.com.

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Sometimes you're faced with challenges that traditional problem solving can't fix. Creative problem solving encourages you to find new, creative ways of thinking that can help you overcome the issue at hand more quickly.

Dialogue mapping is a facilitation technique used to visualize critical thinking as a group. Learn how you and your team can start dialogue mapping today to solve problems and bridge gaps in knowledge and understanding (plus get a free template!).

Bring your bright ideas to life.

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Visual problem solving with flowcharts and mind maps

What’s life without problems? Probably a little boring, if we’re being honest. If everything were perfect all the time, there would be no challenges, and things would get pretty monotonous. This is a rather optimistic view on what many believe to be an aggravating part of life. No matter how you feel about problems, one thing is true: problems are inevitable . You can’t always control how many problems you encounter in your life, but you can learn better ways to solve them. So, what can we do for those really complex issues that aren’t easily solved? Visual problem solving is the perfect way to see solutions and break down complex issues.

Make your own flowchart with Gleek .

What is visual problem solving?

Visual problem solving is the process of using aids like charts or diagrams to display all the aspects of a problem in order to find viable solutions. When problem solving, sometimes it’s hard to see what’s causing the problem, or other relationships and correlations that are affecting whatever it is you’re working on. Two common methods for problem solving include mind maps and flowcharts . A mind map is a non-linear diagram, used for making new ideas or breaking down complex issues. A flowchart is a linear diagram, used for making action plans and describing processes.

5 steps to solve problems

Identify the true problem

Maybe you know what the issue is in clear terms, or perhaps it’s still a little confusing. A good way to get a concrete vision of the problem you need to solve is to pose it as a question, or a short statement. You might come up with something like ‘our sales have dropped’, or, as a question ‘what can we do to increase sales?’.

Get information

Now that you have a clear objective to solve, the next step is to gather all the relevant information that pertains to the issue. This can look like statistics, comments from customers, employee feedback, and more. Once you’ve collected the data, you’ll need to analyze it from all angles to get a clear view on the topic.

Brainstorming session

Get any and all potential solution ideas out on the table. Doesn’t matter how silly an idea seems, just put anything that comes to mind on the drawing board. This is where your visual aids will really come in handy, especially mind maps. You might need more than one chart, depending on how complicated the issue is.

Choose the best idea(s)

Whether on your own or with a team, you’ll have to eliminate the potential solutions that just won’t work. To find the solution that’ll work best, it’s good to analyze it in the same way you did the problem – by looking at potential outcomes, and all facets involved.

Make an action plan

So you think you’ve found the perfect solution! Now what? If your problem is complicated, usually the solution will be too. Here is where another visual aid, like a flowchart, will be helpful. Map out the specific steps you need in order to implement your solution. Then, it’s time to put your plan into action.

These are just the basic steps you can use to start problem solving. You may find that other actions are needed during your own journey.

Common mistakes when problem solving

Mistakes? We all make them from time to time. Here are some common mistakes we are prone to when trying to fix problems.

Undefined problem – When identifying the problem, it’s possible that the problem is too big, multi-faceted, or too complex to tackle all at once. A way to avoid this is to break the problem down into chunks, following common themes.

More problems arise – This isn’t always a direct result of anything we do, but it can happen nonetheless. The best way to deal with more problems that arise when you’re trying to solve the original one is to think of the possible things that could go wrong during the solution stage. When you’re prepared for any situation, you’ll rarely have any setbacks.

No action plan – Finding a way to solve your problem doesn’t mean that the planning is over. On the contrary, you need to create a strategy to properly execute your solution so you won’t end up with a half-solved problem and even more issues than you started with.

When to use flowcharts

One way to chart your problems and progress is through flowcharts. For those who like to think in a step-by-step or linear fashion, flowcharts are the best way to visualize things. Let’s have a look at some situations that are best suited to flowcharts.

Big problems – Flowcharts can help break down a large problem or solution into specific steps or stages from start to finish.

Decision trees – This type of flowchart is helpful when diagramming actions that will happen as a result of other actions, whether they be in a software system or actions taken by people.

Cause and effect – Similar to a decision tree, a cause and effect flowchart is where you can analyze the potential results of various actions, past or present.

Check out our 20 flowchart templates that you can also easily edit !

When to use mind maps

Mind maps are great for brainstorming sessions, and non-linear problem solving. Here are some situations that are best visualized through a mind map.

Finding the problem – So, what is the problem exactly? Sometimes it’s hard to see. Making a mind map offers you the opportunity to see all the moving parts involved with a situation, and how they relate to one another, and can help you suss out the true problem.

Core and branching ideas – You start with a core idea, such as ‘online sales’, then add related ideas or issues branching off from that, like maybe ‘ad revenue’, or ‘social media campaigns’. Then those ideas can have their own branches. This is an easy way to analyze all aspects of a problem.

Source: Problem Solving with Mind Maps (Tutorial)

Looking to create your own flowchart? Gleek has the solution for you. With Gleek, you can create your own flowcharts using a text-based command center, without ever using your mouse. Not only can you create flowcharts, you can create many other UML-based diagrams that will wow your colleagues and bring new life to your presentations. Get started for free today .

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PERSPECTIVE article

Problem-solving is embedded in context… so how do we measure it.

• 1 University of California, Irvine, Irvine, United States
• 2 California State University, Fullerton, Fullerton, California, United States

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Problem solving encompasses the broad domain of human, goal-directed behaviors. Though we may attempt to measure problem solving using tightly controlled and decontextualized tasks, it is inextricably embedded in both reasoners’ experiences and their contexts. Without situating problem-solvers, problem contexts, and our own experiential partialities as researchers, we risk intertwining the research of information relevance with our own confirmatory biases about people, environments, and ourselves. We review each of these ecological facets of information relevance in problem-solving, and we suggest a framework to guide its measurement. We ground this framework with concrete examples of ecologically valid, culturally relevant measurement of problem solving.

Keywords: problem-solving, Measurement, Information relevance, Ecological Validity, Cultural relevance

Received: 01 Feb 2024; Accepted: 26 Apr 2024.

Copyright: © 2024 Rhodes, Richland and Alcalá. 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) or licensor 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: Katherine Rhodes, University of California, Irvine, Irvine, United States

Disclaimer: 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.

Intuition and Visualization in Mathematical Problem Solving

• Published: 09 February 2010
• Volume 29 , pages 29–39, ( 2010 )

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• Valeria Giardino 1  

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In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization.

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Mathematisches Modellieren – Eine Einführung in theoretische und didaktische Hintergründe

For example, at the dawn of set theory, Cantor discusses how to access transfinite numbers. According to him, we get to Cantorian sets operating a double act of abstraction from sets of concrete things. The first act of abstraction brings us to the ‘ordinal number’ or enumeration ; the second act of abstraction brings us to the cardinal number or power of the same set. The cardinal number of M, then, is the general concept that arises from the aggregate M by means of our active faculty of thought. It is thanks to this faculty that we can abstract, and, because of that, provide definitions. By abstracting, we obtain a whole Einheit (“Unity”) of undifferentiated Einsen (“Ones”): according to Cantor, these are ‘objects of our intuition’. See Cantor ( 1915 ).

According to Hadamard, both the preparation and the illumination are mostly subconscious. Nevertheless, he does not deny that conscious thinking is necessary. In fact, once this unconscious illumination has occurred, it must be verified by means of conscious thinking. Intuition allows the mathematician to see the conclusion; then, it is only afterwards that this conclusion will be proved by traditional means. See Hadamard ( 1945 ).

According to Gödel, in physics as well as in logic, we are able to describe, and in fact we do describe, the ultimate reality of things. This happens because we access this nature by means of some immediate capacity: by perception in the case of physics, and by intuition in the case of mathematics. It is mathematical intuition that provides mathematical content. The analogy between perception and intuition can be pushed further. Like perception, intuition is fallible: we can fail in our attempts to get to know the abstract world we are facing. This may mean that further and new intuitions are needed. Therefore, axioms are analogous to physical laws, since it is by means of them that we gain knowledge of the relationships among ‘things’, and we expect experiences to occur in accordance with what these laws prescribe. See Gödel ( 1986 ).

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Acknowledgments

I thank Roberto Casati, Davide Crippa, Leon Horsten, John Mumma, and Mario Piazza for their useful suggestions on a first draft of the article. The research was supported by the European Community’s Seventh Framework Program ([FP7/2007-2013] under a Marie Curie Intra-European Fellowship for Career Development, contract number no. 220686—DBR (Diagram-based Reasoning).

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Valeria Giardino

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Giardino, V. Intuition and Visualization in Mathematical Problem Solving. Topoi 29 , 29–39 (2010). https://doi.org/10.1007/s11245-009-9064-5

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Published : 09 February 2010

Issue Date : April 2010

DOI : https://doi.org/10.1007/s11245-009-9064-5

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Traveling Salesman Problem with Python: Greedy and Brute Force

Just imagine yourself as a delivery courier. You have to deliver multiple parcels to many different spots. You also aim to minimize the fuel costs and time of traveling to maximize profit. This generally creates confusion about where to deliver parcels first and which routes to take.

In this article, we will learn about the famous problem of Travelling Salesman. We will solve this issue using Python programming language. We will also try to plot the best route to be taken and calculate the minimum distance to be traveled.

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Solving the Traveling Salesman Problem with Python

The problem statement is that a salesman has to travel to the Indian cities of Mumbai, Delhi, Bangalore, Hyderabad, Ahmedabad, Chennai, Kolkata, Surat, Pune, and Jaipur to sell some products.

The objective of the problem is to minimize the total distance travelled by the salesman.

This describes our problem statement. Now let’s move on to how to solve this problem using Python programming language.

We will look at two ways of solving the problem – using the Greedy algorithm and the Brute Force method ( this method is the most preferred one as it provides more optimal solutions ).

Example Usage and Visualization of the Greedy Algorithm

Let us look at the code of the Simple Greedy algorithm.

In the above example, we randomly input some coordinates and calculated the best route. Let us look at the output of this method.

Thus, according to the Greedy algorithm, we will travel to city 1 first, and cities 1,3,2 after that.

Brute Force Approach to the Traveling Salesman Problem

Let us look at the code of the Brute Force Method.

Let us look at the output of the Brute Force method.

Thus, according to the Brute force method, the minimum distance is approximately 230 units.

Greedy Algorithm vs. Brute Force Method

The difference between the Greedy algorithm and the Brute Force method is that the Greedy algorithm builds up the solution step by step whereas in the Brute Force method, all the permutations of the solution are found and then the solution with minimum distance is selected.

Here you go! Now you know the Travelling Salesman Problem and how to solve it using Python. In this article, you learned about two methods of approach i.e. Greedy algorithm and Travelling Salesman problem.

Hope you enjoyed reading it!

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Computer Science > Neural and Evolutionary Computing

Title: an efficient reconstructed differential evolution variant by some of the current state-of-the-art strategies for solving single objective bound constrained problems.

Abstract: Complex single-objective bounded problems are often difficult to solve. In evolutionary computation methods, since the proposal of differential evolution algorithm in 1997, it has been widely studied and developed due to its simplicity and efficiency. These developments include various adaptive strategies, operator improvements, and the introduction of other search methods. After 2014, research based on LSHADE has also been widely studied by researchers. However, although recently proposed improvement strategies have shown superiority over their previous generation's first performance, adding all new strategies may not necessarily bring the strongest performance. Therefore, we recombine some effective advances based on advanced differential evolution variants in recent years and finally determine an effective combination scheme to further promote the performance of differential evolution. In this paper, we propose a strategy recombination and reconstruction differential evolution algorithm called reconstructed differential evolution (RDE) to solve single-objective bounded optimization problems. Based on the benchmark suite of the 2024 IEEE Congress on Evolutionary Computation (CEC2024), we tested RDE and several other advanced differential evolution variants. The experimental results show that RDE has superior performance in solving complex optimization problems.

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