applications and problem solving

What Is Problem Solving? How Software Engineers Approach Complex Challenges

From debugging an existing system to designing an entirely new software application, a day in the life of a software engineer is filled with various challenges and complexities. The one skill that glues these disparate tasks together and makes them manageable? Problem solving . 

Throughout this blog post, we’ll explore why problem-solving skills are so critical for software engineers, delve into the techniques they use to address complex challenges, and discuss how hiring managers can identify these skills during the hiring process. 

What Is Problem Solving?

But what exactly is problem solving in the context of software engineering? How does it work, and why is it so important?

Problem solving, in the simplest terms, is the process of identifying a problem, analyzing it, and finding the most effective solution to overcome it. For software engineers, this process is deeply embedded in their daily workflow. It could be something as simple as figuring out why a piece of code isn’t working as expected, or something as complex as designing the architecture for a new software system. 

In a world where technology is evolving at a blistering pace, the complexity and volume of problems that software engineers face are also growing. As such, the ability to tackle these issues head-on and find innovative solutions is not only a handy skill — it’s a necessity. 

The Importance of Problem-Solving Skills for Software Engineers

Problem-solving isn’t just another ability that software engineers pull out of their toolkits when they encounter a bug or a system failure. It’s a constant, ongoing process that’s intrinsic to every aspect of their work. Let’s break down why this skill is so critical.

Driving Development Forward

Without problem solving, software development would hit a standstill. Every new feature, every optimization, and every bug fix is a problem that needs solving. Whether it’s a performance issue that needs diagnosing or a user interface that needs improving, the capacity to tackle and solve these problems is what keeps the wheels of development turning.

It’s estimated that 60% of software development lifecycle costs are related to maintenance tasks, including debugging and problem solving. This highlights how pivotal this skill is to the everyday functioning and advancement of software systems.

Innovation and Optimization

The importance of problem solving isn’t confined to reactive scenarios; it also plays a major role in proactive, innovative initiatives . Software engineers often need to think outside the box to come up with creative solutions, whether it’s optimizing an algorithm to run faster or designing a new feature to meet customer needs. These are all forms of problem solving.

Consider the development of the modern smartphone. It wasn’t born out of a pre-existing issue but was a solution to a problem people didn’t realize they had — a device that combined communication, entertainment, and productivity into one handheld tool.

Increasing Efficiency and Productivity

Good problem-solving skills can save a lot of time and resources. Effective problem-solvers are adept at dissecting an issue to understand its root cause, thus reducing the time spent on trial and error. This efficiency means projects move faster, releases happen sooner, and businesses stay ahead of their competition.

Improving Software Quality

Problem solving also plays a significant role in enhancing the quality of the end product. By tackling the root causes of bugs and system failures, software engineers can deliver reliable, high-performing software. This is critical because, according to the Consortium for Information and Software Quality, poor quality software in the U.S. in 2022 cost at least $2.41 trillion in operational issues, wasted developer time, and other related problems.

Problem-Solving Techniques in Software Engineering

So how do software engineers go about tackling these complex challenges? Let’s explore some of the key problem-solving techniques, theories, and processes they commonly use.

Decomposition

Breaking down a problem into smaller, manageable parts is one of the first steps in the problem-solving process. It’s like dealing with a complicated puzzle. You don’t try to solve it all at once. Instead, you separate the pieces, group them based on similarities, and then start working on the smaller sets. This method allows software engineers to handle complex issues without being overwhelmed and makes it easier to identify where things might be going wrong.

Abstraction

In the realm of software engineering, abstraction means focusing on the necessary information only and ignoring irrelevant details. It is a way of simplifying complex systems to make them easier to understand and manage. For instance, a software engineer might ignore the details of how a database works to focus on the information it holds and how to retrieve or modify that information.

Algorithmic Thinking

At its core, software engineering is about creating algorithms — step-by-step procedures to solve a problem or accomplish a goal. Algorithmic thinking involves conceiving and expressing these procedures clearly and accurately and viewing every problem through an algorithmic lens. A well-designed algorithm not only solves the problem at hand but also does so efficiently, saving computational resources.

Parallel Thinking

Parallel thinking is a structured process where team members think in the same direction at the same time, allowing for more organized discussion and collaboration. It’s an approach popularized by Edward de Bono with the “ Six Thinking Hats ” technique, where each “hat” represents a different style of thinking.

In the context of software engineering, parallel thinking can be highly effective for problem solving. For instance, when dealing with a complex issue, the team can use the “White Hat” to focus solely on the data and facts about the problem, then the “Black Hat” to consider potential problems with a proposed solution, and so on. This structured approach can lead to more comprehensive analysis and more effective solutions, and it ensures that everyone’s perspectives are considered.

This is the process of identifying and fixing errors in code . Debugging involves carefully reviewing the code, reproducing and analyzing the error, and then making necessary modifications to rectify the problem. It’s a key part of maintaining and improving software quality.

Testing and Validation

Testing is an essential part of problem solving in software engineering. Engineers use a variety of tests to verify that their code works as expected and to uncover any potential issues. These range from unit tests that check individual components of the code to integration tests that ensure the pieces work well together. Validation, on the other hand, ensures that the solution not only works but also fulfills the intended requirements and objectives.

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Evaluating Problem-Solving Skills

We’ve examined the importance of problem-solving in the work of a software engineer and explored various techniques software engineers employ to approach complex challenges. Now, let’s delve into how hiring teams can identify and evaluate problem-solving skills during the hiring process.

Recognizing Problem-Solving Skills in Candidates

How can you tell if a candidate is a good problem solver? Look for these indicators:

• Previous Experience: A history of dealing with complex, challenging projects is often a good sign. Ask the candidate to discuss a difficult problem they faced in a previous role and how they solved it.
• Problem-Solving Questions: During interviews, pose hypothetical scenarios or present real problems your company has faced. Ask candidates to explain how they would tackle these issues. You’re not just looking for a correct solution but the thought process that led them there.
• Technical Tests: Coding challenges and other technical tests can provide insight into a candidate’s problem-solving abilities. Consider leveraging a platform for assessing these skills in a realistic, job-related context.

Assessing Problem-Solving Skills

Once you’ve identified potential problem solvers, here are a few ways you can assess their skills:

• Solution Effectiveness: Did the candidate solve the problem? How efficient and effective is their solution?
• Approach and Process: Go beyond whether or not they solved the problem and examine how they arrived at their solution. Did they break the problem down into manageable parts? Did they consider different perspectives and possibilities?
• Communication: A good problem solver can explain their thought process clearly. Can the candidate effectively communicate how they arrived at their solution and why they chose it?
• Adaptability: Problem-solving often involves a degree of trial and error. How does the candidate handle roadblocks? Do they adapt their approach based on new information or feedback?

Hiring managers play a crucial role in identifying and fostering problem-solving skills within their teams. By focusing on these abilities during the hiring process, companies can build teams that are more capable, innovative, and resilient.

Key Takeaways

As you can see, problem solving plays a pivotal role in software engineering. Far from being an occasional requirement, it is the lifeblood that drives development forward, catalyzes innovation, and delivers of quality software. 

By leveraging problem-solving techniques, software engineers employ a powerful suite of strategies to overcome complex challenges. But mastering these techniques isn’t simple feat. It requires a learning mindset, regular practice, collaboration, reflective thinking, resilience, and a commitment to staying updated with industry trends. 

For hiring managers and team leads, recognizing these skills and fostering a culture that values and nurtures problem solving is key. It’s this emphasis on problem solving that can differentiate an average team from a high-performing one and an ordinary product from an industry-leading one.

At the end of the day, software engineering is fundamentally about solving problems — problems that matter to businesses, to users, and to the wider society. And it’s the proficient problem solvers who stand at the forefront of this dynamic field, turning challenges into opportunities, and ideas into reality.

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Problem-Solving Strategies and Obstacles

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

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

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

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35 problem-solving techniques and methods for solving complex problems

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All teams and organizations encounter challenges as they grow. There are problems that might occur for teams when it comes to miscommunication or resolving business-critical issues . You may face challenges around growth , design , user engagement, and even team culture and happiness. In short, problem-solving techniques should be part of every team’s skillset.

Problem-solving methods are primarily designed to help a group or team through a process of first identifying problems and challenges , ideating possible solutions , and then evaluating the most suitable .

Finding effective solutions to complex problems isn’t easy, but by using the right process and techniques, you can help your team be more efficient in the process.

So how do you develop strategies that are engaging, and empower your team to solve problems effectively?

In this blog post, we share a series of problem-solving tools you can use in your next workshop or team meeting. You’ll also find some tips for facilitating the process and how to enable others to solve complex problems.

Let’s get started! 

How do you identify problems?

How do you identify the right solution.

• Tips for more effective problem-solving

Complete problem-solving methods

• Problem-solving techniques to identify and analyze problems
• Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve. 

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward. 

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.  

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.  

Try different approaches  

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally 

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator 

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

• Six Thinking Hats
• Lightning Decision Jam
• Problem Definition Process
• Discovery & Action Dialogue

Design Sprint 2.0

• Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on. 

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages. 

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design. 

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition   #problem solving   #idea generation   #creativity   #online   #remote-friendly   A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys 

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges. 

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results. 

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys   #hyperisland   #innovation   This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold. 

World Cafe   #hyperisland   #innovation   #issue analysis   World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

6. Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD)   #idea generation   #liberating structures   #action   #issue analysis   #remote-friendly   DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology   #action plan   #idea generation   #problem solving   #issue analysis   #large group   #online   #remote-friendly   Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

• The Creativity Dice
• Fishbone Analysis
• Problem Tree
• SWOT Analysis
• Agreement-Certainty Matrix
• The Journalistic Six
• LEGO Challenge
• What, So What, Now What?
• Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed. 

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.  

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It!   #gamestorming   #problem solving   #action   Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed. 

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable. 

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around. 

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish. 

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis   #problem solving   ##root cause analysis   #decision making   #online facilitation   A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree 

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them. 

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree   #define intentions   #create   #design   #issue analysis   A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward. 

SWOT analysis   #gamestorming   #problem solving   #action   #meeting facilitation   The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results. 

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process. 

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID   #gamestorming   #project planning   #issue analysis   #problem solving   When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!   

Speed Boat   #gamestorming   #problem solving   #action   Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How   #idea generation   #issue analysis   #problem solving   #online   #creative thinking   #remote-friendly   A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills. 

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO! 

LEGO Challenge   #hyperisland   #team   A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems. 

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken. 

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What?   #issue analysis   #innovation   #liberating structures   You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists  

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists   #vision   #big picture   #issue analysis   #remote-friendly   This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions 

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

• Improved Solutions
• Four-Step Sketch
• 15% Solutions
• How-Now-Wow matrix
• Impact Effort Matrix

21. Mindspin  

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly. 

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation. 

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex. 

MindSpin   #teampedia   #idea generation   #problem solving   #action   A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result. 

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution. 

Improved Solutions   #creativity   #thiagi   #problem solving   #action   #team   You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged. 

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch   #design sprint   #innovation   #idea generation   #remote-friendly   The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper,  Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change. 

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.   

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change. 

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process. 

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud. 

How-Now-Wow Matrix   #gamestorming   #idea generation   #remote-friendly   When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice. 

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them. 

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action? 

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus. 

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively. 

Dotmocracy   #action   #decision making   #group prioritization   #hyperisland   #remote-friendly   Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

• Check-in/Check-out
• Doodling Together
• Show and Tell
• Constellations
• Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute. 

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out   #team   #opening   #closing   #hyperisland   #remote-friendly   Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together  

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start. 

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems. 

Doodling Together   #collaboration   #creativity   #teamwork   #fun   #team   #visual methods   #energiser   #icebreaker   #remote-friendly   Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team! 

Show and Tell   #gamestorming   #action   #opening   #meeting facilitation   Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible. 

Constellations   #trust   #connection   #opening   #coaching   #patterns   #system   Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic. 

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree   #thiagi   #opening   #perspectives   #remote-friendly   With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

• One Breath Feedback
• Who What When Matrix
• Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space. 

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round. 

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them. 

One breath feedback   #closing   #feedback   #action   This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix 

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward. 

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved. 

Who/What/When Matrix   #gamestorming   #action   #project planning   With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out! 

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised. 

Response Cards   #debriefing   #closing   #structured sharing   #questions and answers   #thiagi   #action   It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks  to build your agenda. When you make changes or update your agenda, your session  timing   adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore  how to use SessionLab  to design effective problem solving workshops or  watch this five minute video  to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you! 

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

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8D Problem Solving: Comprehensive Breakdown and Practical Applications

The 8D problem-solving process stands as a beacon of structured analysis and corrective action within the complexities of operational pitfalls and quality control discrepancies across industries. Originating from the automotive industry and since adopted widely, the methodology offers a meticulous step-by-step approach that fosters team cohesion, addresses problems at their roots, and implements sustainable solutions.

This article seeks to delve into the nuances of the 8D problem-solving framework, presenting a lucid exposition of its origins, a detailed foray into each step enriched by practical examples, and concluding with the unequivocal benefit bouquet it presents to the organization adopting it.

The Origins of the 8D Problem Solving Methodology

The 8D, or "Eight Disciplines," problem-solving approach germinated from the fertile grounds of collaborative efforts to ensure superior quality and reliability in manufacturing. Initially developed by the Ford Motor Company in the 1980s, this systematic method was a response to a confluence of quality and operational issues that were pervasive in the automotive industry. It drew broader appeal as its efficacy became apparent - functioning as an amalgam of logic, analytics, and teamwork to tackle problems methodically.

The wide reach of the 8D methodology is evident in industries ranging from manufacturing to healthcare, aerospace to IT, and beyond. Its universal applicability stems from a foundational adherence to principle over process, transcending the intricacies of industry-specific challenges. By combining reactive and proactive measures, the 8D method helps in not just extinguishing the fire, but also preventing its outbreak, making it an enduring asset in the organizational toolkit.

The 8 Steps of Problem Solving

An incursion into the 8D methodology reveals a framework that is both systematic and flexible. Each step is sequenced to ensure that issues are not merely patched but genuinely resolved, implementing robust preventive measures to curtail recurrences. This section expounds on each disciplinary step and serves as a substrate for practical implementation examples, supplementing theoretical insights with real-world applicability.

Step 1: Establish a Team

The cornerstone of any formidable 8D approach begins with assembling a competent team. The wisdom embedded in this initial phase is the recognition that effective problem-solving is not a solitary venture but a collaborative pursuit. A multidisciplinary team brings diverse perspectives that are critical in diagnosing issues accurately and devising solutions effectively.

When determining team composition, the emphasis should be on a mix of skills and expertise relevant to the problem at hand. Roles within the team should be clearly defined to streamline activities and foster accountability. Each member should be well-versed in their responsibilities, from those leading the problem-solving charge to those executing and tracking actions.

Step 2: Describe the Problem

Clarity is vital in the second step, which necessitates delineating the problem with precision. A meticulous description sets the foundation for targeted analysis and actionable solutions. It involves accruing information that is factual, quantifiable, and devoid of assumptions – the cornerstone of an accurate problem portrayal.

Techniques like '5W2H' (who, what, when, where, why, how, how much) can galvanize teams into crafting detailed problem descriptions. An exemplar of a well-articulated problem statement might state, "Machine X has experienced a 15% decline in output quality, resulting in a monthly loss of 200 units of product Y since January due to recurrent mechanical inaccuracies."

Step 3: Develop Interim Containment Actions

Addressing a problem's immediate impact is pivotal to prevent exacerbation as a root cause analysis is conducted. Interim containment actions can be likened to first aid – essential, though not the definitive cure. These measures should be rigorously designed to quell the problem's spread or intensification without creating new issues.

An interim action for the aforementioned issue with Machine X could involve adjusting the production schedule to mitigate output loss while the machine is under examination. This demonstrates a temperate solution, buying time for a comprehensive fix without severely disrupting the production chain.

Step 4: Define and Verify the Root Cause(s)

Singular in its focus yet pluralistic in its approach, this phase is committed to uncovering the underlying reasons for the problem. Root cause identification is a task of surgical precision, necessitating a deep dive into the problem without the constraints of predetermined notions.

Techniques such as the "5 Whys" and "Fishbone Diagram" guide problem solvers through a structured investigation of potential causes. Verification is as crucial as identification, ensuring that purported root causes stand up to scrutiny and testing.

Step 5: Verify Permanent Corrective Action(s)

Once root causes have been established, attention shifts to devising and validating long-term corrective actions. This step traverses the path from theory to practice. It requires a judicious appraisal of potential solutions with a clear-eyed view of their feasibility, effectiveness, and sustainability.

Best practices in this step incorporate piloting solutions on a smaller scale, enabling refinement before full-scale implementation. A well-considered corrective action for Machine X might involve upgrading mechanical components identified as failure points, subject to cost-benefit analysis and potential disruption to the production line.

Step 6: Implementing and Validating Permanent Corrective Actions

This step transitions the plan into reality, pushing the corrective actions beyond the threshold into the operational environment. Careful implementation is the linchpin, with detailed plans and schedules ensuring that actions are well-executed and efficacious.

The validation process is a keystone in affirming that corrective actions deliver the intended improvements. For Machine X, this could entail monitoring post-repair performance metrics over a defined period against pre-issue levels to authenticate the efficacy of the improvements.

Step 7: Preventive Measures

Armed with insights gleaned, the 7th step propels the methodology into preventative mode. Here, the onus is on forestalling a problem’s resurgence by ingraining the lessons learned into the organizational fabric. The aim is to encapsulate these insights in policies, procedures, or system changes.

This could mean revising maintenance schedules or worker training programs for Machine X to include the specific nuances that led to the mechanical inaccuracies, thereby shielding against repeat episodes.

Step 8: Congratulate Your Team

The final step encompasses a human-centered focus on recognition and commendation. Acknowledgment of the team’s efforts reinforces motivation, fosters a positive culture, and encourages engagement in future problem-solving initiatives.

Celebrating the success could manifest in a ceremonious recognition of the team’s achievements, an internal announcement of their contributions, or a tangible expression of appreciation. This not only cements the accomplishment but also propels a sense of camaraderie and collective purpose.

The Importance of the 8D Problem Solving Process

A mature consideration of the 8D problem-solving process corroborates its contributory significance in unraveling complex issues and instituting consequential improvements. The benefits it confers are manifest in enhanced product quality, heightened customer satisfaction, and the stimulation of a proactive problem-solving culture. Challenges do persist, mainly in the form of resistance to change or insufficient training; nevertheless, with a conscientious implementation, these can be navigated.

Moreover, the 8D approach aligns seamlessly with the pursuit of continuous improvement – a cornerstone of many business philosophies such as Lean and Six Sigma. It thus serves not only as a solution framework but also as a catalyst for organizational growth and learning.

In summary, the 8D problem-solving methodology stands out for its disciplined, team-driven, and methodical approach to tackling complex problems. From its historical roots in the automotive industry to its implementation in modern enterprises, its efficacy in achieving sustainable solutions is undoubted. Online certificate programs and problem-solving courses often feature 8D due to its relevance and value across industries.

As this article delineates each step, with practical applications and advice, the message is clear: mastery of 8D is not just for immediate problem resolution – it is a pathway to building a resilient and adaptive organization capable of facing the challenges of an ever-changing business landscape.

What are the key steps involved in the 8D Problem Solving process and how do they interact with each other?

Introduction to the 8d process.

The 8D Problem Solving process stands tall. It is a structured approach. Businesses use it widely. 8D tackles complex problems effectively. It drives teams towards lasting solutions. It also fosters quality and reliability. The "D" denotes eight disciplined steps. These steps guide teams. They identify, correct, and prevent issues.

8D Steps Explained

D1: establish the team.

Form a skilled team first. Diversity matters here. Each member brings insights. Their combined expertise is crucial. Team formation kicks off the process.

D2: Describe the Problem

Articulate the issue clearly. Use quantifiable data here. Understanding the scope matters. Have accurate problem statements ready. They pave the way forward.

D3: Develop Interim Containment Action

Ensure a temporary fix. It limits the problem's impact. Speed is of the essence. However, ensure the action is effective. The goal is to stabilize the situation.

D4: Determine Root Cause

Dig deep into causes. Use data-driven analysis. Techniques include fishbone diagrams. Five Whys is also popular. Root cause analysis is pivotal. It prepares the team for corrective actions.

D5: Design and Verify Permanent Corrective Actions

Choose the best corrective action. Rigorous selection criteria apply. Effectiveness and efficiency matter. Verify through testing. Make certain the solution fits.

D6: Implement and Validate Permanent Corrective Actions

Roll out the solution. Watch closely for results. Validation takes place here. Use performance indicators for this. They must indicate that the solution works.

D7: Prevent Recurrence

Embed the improvement. Update systems and policies. Training may be necessary. Maintain the gains. This step safeguards the future. Documentation is important here.

D8: Congratulate the Team

Never overlook recognition. Acknowledge everyone's efforts. Celebrate the success achieved. It boosts team morale. It also promotes a culture of quality.

Interplay Between Steps

The interdependence is strong. Each step builds on the previous one. For example, a strong team in D1 enhances problem understanding in D2. Similarly, effective interim actions in D3 set the stage for a thorough root cause analysis in D4.

The verification in D5 ensures the solution from D4 is sound. Implementation in D6 then relies on the verified action. To prevent recurrence (D7), one must understand the root cause. The entire process relies on clear communication. Team recognition (D8) closes the loop neatly. It paves the way for future problem-solving success.

In essence, the 8D steps are interlinked. Each step informs the next. Teams achieve the best results by following the sequence. They also adapt as needed. 8D enforces a discipline that leads to high-quality results. The interaction between steps ensures problems do not just disappear. They stay solved. This is the power of an integrated problem-solving approach.

Can you provide some practical examples of the effective application of 8D Problem Solving strategies in real-life settings?

Understanding 8d problem solving.

8D problem solving stands for Eight Disciplines. It involves steps that teams must follow. Starting from identifying problems , it goes until preventing reoccurrences . Companies use 8D to tackle complex issues. Its main goal remains quality improvement.

Here are practical examples where 8D shines.

Example 1: Automotive Industry

D0: Plan - An auto manufacturer formed a team. Their goal was clear: resolve brake failures.

D1: Team Formation - They gathered experts from diverse fields. Collaboration was key here.

D2: Describe the Problem - They identified specific issues. Customers reported brakes failing at high speeds.

D3: Develop Interim Containment - They distributed quick-fix kits to dealerships. This ensured immediate customer safety.

D4: Determine Root Causes - A deep dive ensued. The team discovered a faulty brake fluid line.

D5: Choose and Verify Permanent Corrective Actions (PCAs) - They redesigned the brake line. Then they tested it under rigorous conditions.

D6: Implement and Validate PCAs - The new design went into production. Ongoing assessments confirmed its effectiveness.

D7: Take Preventive Measures - They updated their design guidelines. Thus, they eliminated the possibility of similar failures.

D8: Congratulate Your Team - Management recognized the team's effort. This promoted a culture of problem-solving.

Example 2: Electronics Manufacturer

D0: Plan - A sudden surge in returned gadgets prompted action.

D1: Team Formation - A cross-functional team took charge. They had one aim: find the flaw.

D2: Describe the Problem - Devices were overheating during usage. Anxiety among customers grew.

D3: Develop Interim Containment - They halted the production line. Assessing risks was necessary.

D4: Determine Root Causes - Detailed analysis revealed a substandard battery component.

D5: Choose and Verify PCAs - They sourced a higher quality component. Subsequent tests showed promising results.

D6: Implement and Validate PCAs - They integrated the new component into production. Monitoring phases ensured it was a fix.

D7: Take Preventive Measures - They revamped their quality control protocols. Now they could avoid similar issues.

D8: Congratulate Your Team - The team's innovative approach earned praise. They set new standards in their processes.

Example 3: Food Packaging Company

D0: Plan - Reports of packaging leaks triggered an 8D.

D1: Team Formation - Experts from production to distribution joined forces. They understood the stakes were high.

D2: Describe the Problem - The leaks were sporadic but damaging. Food safety concerns escalated.

D3: Develop Interim Containment - They removed compromised products from shelves. Protecting the consumer was paramount.

D4: Determine Root Causes - Investigation exposed a sealing machine defect.

D5: Choose and Verify PCAs - Engineers redesigned the sealing mechanism. Trials followed, proving success.

D6: Implement and Validate PCAs - The updated machines replaced the old ones. Continuous evaluations followed to assure quality.

D7: Take Preventive Measures - They introduced more rigorous maintenance routines. They aimed to preempt future failures.

D8: Congratulate Your Team - The swift and thorough response earned accolades. They reinforced trust in their brand.

8D's Practical Value

Each example showcases 8D's potential. This problem-solving framework adapts to various scenarios. Through structured teamwork and analysis, it guides toward sustainable solutions. It helps in ensuring the same problem does not occur twice. Businesses across different sectors find 8D crucial for their continuous improvement efforts. It underlines that a methodical approach to problem-solving can yield significant long-term benefits.

How is the effectiveness and success of the 8D Problem Solving approach measured in practical applications?

Introduction to 8d problem solving.

The 8D Problem Solving approach stands as a structured methodology. It aims to address and resolve critical issues within an organization. Rooted in the team-oriented approach, 8D follows eight disciplined steps. These steps ensure a comprehensive and effective resolution process. The process includes identifying the problem, implementing interim controls, defining root causes, developing a corrective action plan, taking corrective actions, validating those actions, preventing recurrence, and finally congratulating the team.

Measuring Effectiveness and Success

Quantitative metrics.

Timeliness of Response

The promptness of the initial response is critical. It alerts stakeholders to the emergence and acknowledgment of the issue.

Problem Recurrence Rates

A key success indicator is the frequency of problem recurrence. A declining trend signifies effective corrective actions.

Financial Impact

Cost savings or avoidance measures the fiscal efficiency of the resolution. It counts both direct and indirect factors.

Cycle Time Reduction

Improvements in processes can lead to shorter cycle times. This reflects efficiency gains from the 8D implementation.

Qualitative Metrics

Quality of Documentation

Comprehensive documentation ensures thorough issue analysis. It captures the nuances of the problem-solving journey.

Stakeholder Satisfaction

Feedback from affected parties gauges the outcome’s acceptability. Satisfaction levels can direct future interventions.

Knowledge Transfer

Disseminating learnings enhances organizational capability. Sharing insights leads to broader, preventive measures.

Team Cohesion and Growth

Personal and team development signal process benefits. Such growth provides intangible value to the organization.

Practical Application and Continuous Improvement

In practical applications, tailoring metrics to contexts is vital. Unique business environments demand specific success criteria. Therefore, adapting the approach and its measurement system is necessary.

Organizations may employ a combination of tangible and intangible metrics. Aligning these to strategic goals ensures relevance. The 8D Process receives fine-tuning through iterative cycles. Each cycle offers an opportunity for enhanced problem-solving efficacy.

The Importance of Measure Standardization

Standardizing the measurement process ensures consistency. It aids in comparing and benchmarking against best practices. Homogeneity in measures facilitates clearer communication. It enhances the understanding of successes and areas for improvement.

Revisiting and Refining the 8D Process

Upon completion, a rigorous review of the 8D process is crucial. It ensures learnings lead to process refinement. Alterations in measures might follow to better reflect evolving business needs. This ongoing evolution drives the sustained value of the 8D methodology.

The 8D Problem Solving approach, with its disciplined steps, delivers a robust framework. Measuring its effectiveness requires a blend of quantitative and qualitative metrics. These metrics, when standardized and continually refined, offer a clear lens to assess the 8D process's success. They help organizations not just to solve problems but to evolve in their problem-solving capabilities.

He is a content producer who specializes in blog content. He has a master's degree in business administration and he lives in the Netherlands.

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Module 2: General Problem Solving

Discussion: problem solving application.

Pick a unique real problem and try to solve it using the general problem solving strategies from this module. Present the problem and the solution to the rest of the class. View the problems posted by your classmates and respond to at least two.

This assignment will help you practice applying what you have learned about general problem solving strategies to real problems or situations, and communicating about mathematical ideas with others.

Pick a problem:

Option 1:   Pick a real problem that you need to solve and use the different general problem solving strategies to try to find a solution.   For example, perhaps you need to figure out how to pack an awkward item in your car trunk, or decide what to spend the most time studying for an exam in another class, or plan the assignments and schedule for a charity clean-up project you are organizing.

Option 2:   Pick a real problem that someone else has solved and describe the general strategies they used.   This could be a friend or relative who solved a problem like the ones described in Option 1, or it could be a historical person, like George Washington Carver, who invented peanut butter and many other things, or Mary Anderson, who invented the windshield wiper.

Option 3:   Pick a currently unsolved problem, like curing cancer or finding better energy sources, and analyze how people are using the different problem solving strategies to find a solution.

Solve the problem or analyze the solution:

If you are solving your own problem, use the different general problem solving strategies that you learned about in this unit to try to solve it.   If you are analyzing a problem someone else solved, or is trying to solve, identify the different general problem solving strategies used.

Present the problem, the general problem solving strategies used, and the solution to the others in the class:

Post a message in the application discussion forum for this unit.   In your message, describe the problem and how you solved it.   Use the equation editor as necessary to show any mathematical operations.   The better you communicate, the more points you will earn.   If you enjoy and know how to use multimedia, such as video, audio, and graphics, you may use those as well, but this is not required.

View and respond to the application problems submitted by your classmates.

Pick two of your classmates’ applications that were particularly helpful to you.   Write a response to each, explaining in a paragraph or two why their applications helped you better understand the mathematics for this unit or better understand how the mathematics for this unit could be used outside of class.

Background: This Saturday is the annual chocolate chip cookie bake sale event. At each of the last 2 bake sale events, there were 600 people who attended. Every annual bake sale must receive 600 attendees. Every attendee eats 5 cookies each.

Problem: We must raise $1,500 for our daughter’s team travel expenses. How many cookies must we make and how much money must each cookie sell for in order to raise the appropriate money?

Solution: 1) The amount of cookies to make are 600 (attendees) times 5 (cookies eaten per person) equals 3,000 cookies to make. 2) 3,000 (cookies made) divided by $1,500 (money needed) equal 50 cents per cookie.

Other Topics:   shipping textbooks, pattern recognition, MPG, budgeting, transportation, metric/English conversions, lottery, time management, retail sales, volume of dirt on a softball field, tipping at a restaurant, yard sales, auto part manufacturing, room design, rate of change, estimate the square root of 3 to two decimal places with a calculator without square root key

• Mathematics for the Liberal Arts I. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : http://eli.nvcc.edu/ . License : CC BY: Attribution

Decoding AI’s Problem-Solving Capability: A Comprehensive Guide

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In the vast and evolving landscape of Artificial Intelligence (AI), the problem-solving capability of AI stands as a cornerstone, showcasing the remarkable ability of machines to mimic human-like decision-making and creativity. This problem-solving capability enables AI to analyze complex scenarios, identify patterns, and devise effective solutions, often surpassing human speed and accuracy. But what exactly encompasses the problem-solving capability within the context of AI, and how does it operate?

Our exploration delves into the mechanisms behind AI’s problem-solving capability, tackling everything from simple puzzles to complex, real-world challenges. By demystifying the problem-solving capability of AI, we aim to provide a clearer understanding of this fascinating field, making it accessible and engaging for college students and tech enthusiasts alike. Prepare to embark on a journey into the heart of AI, where innovation meets practicality in harnessing AI’s problem-solving capability to solve the unsolvable.

• What is Problem-Solving Capability in AI?
• The Mechanisms Behind AI Problem-Solving
• Types of Problems AI Can Solve
• Techniques AI Uses to Solve Problems
• Real-World Applications of AI Problem-Solving
• Challenges in AI’s Problem-Solving Capabilities
• The Future of AI Problem-Solving

1. What is Problem-Solving Capability in AI?

Problem-solving capability in Artificial Intelligence refers to the ability of AI systems to identify, analyze, and solve problems autonomously. This involves understanding the problem at hand, breaking it down into manageable components, and applying logical strategies to arrive at a solution. Unlike traditional computing that follows predefined paths, AI problem-solving encompasses learning from data, adapting to new situations, and making decisions with minimal human intervention.

At its core, AI problem-solving is grounded in the field of cognitive science, which studies how human thought processes are replicated by machines. This capability is not just about finding any solution but about identifying the most efficient and effective solution among many possibilities. It leverages a combination of algorithms, models, and data to mimic the human ability to reason, learn from experience, and apply knowledge to new and unseen scenarios.

AI problem-solving capabilities span various domains, from simple tasks like solving puzzles to complex decisions in financial analysis, healthcare diagnostics, and beyond. These capabilities are powered by different branches of AI, including machine learning, deep learning, natural language processing, and robotics, each contributing to the AI’s ability to tackle specific types of problems.

2. The Mechanisms Behind AI Problem-Solving

AI’s ability to solve problems hinges on several key mechanisms, each contributing to the system’s overall intelligence and functionality. Understanding these mechanisms provides insight into how AI navigates complex challenges:

• Data Processing and Pattern Recognition: At the heart of AI problem-solving is the ability to process vast amounts of data, identifying patterns and insights that are not immediately apparent. Through techniques like machine learning, AI systems learn from data, improving their problem-solving capabilities over time.
• Algorithmic Efficiency: AI relies on sophisticated algorithms that enable it to search through potential solutions quickly and effectively. These algorithms, ranging from simple decision trees to complex neural networks, are designed to optimize the search process, reducing the time and resources required to find a solution.
• Heuristic Techniques: AI often employs heuristics, or rules of thumb, to streamline the problem-solving process. By making educated guesses, AI can bypass unnecessary calculations, focusing on the most promising paths to a solution.
• Adaptability and Learning: A defining feature of AI’s problem-solving capability is its ability to adapt and learn from experience. Through techniques like reinforcement learning, AI systems refine their strategies based on feedback, becoming more efficient problem solvers over time.
• Simulated Annealing and Genetic Algorithms: For particularly complex problems, AI uses advanced strategies like simulated annealing and genetic algorithms, which mimic natural processes to explore a wide range of potential solutions, gradually honing in on the most effective ones.

3. Types of Problems AI Can Solve

AI’s problem-solving capabilities are not limited to a single domain but span across various fields, demonstrating its versatility and power:

• Logical Problems: AI excels at solving logical puzzles and games, such as chess and Go, where success depends on strategic planning and the ability to anticipate opponents’ moves.
• Predictive Modeling: In fields like finance and weather forecasting, AI analyzes historical data to make accurate predictions about future events, helping experts make informed decisions.

• Natural Language Understanding: AI tackles the challenge of understanding human language, enabling applications like chatbots and voice assistants to interpret and respond to user requests accurately.
• Image and Pattern Recognition: From medical diagnostics to autonomous vehicles, AI’s ability to recognize patterns in images enables it to identify diseases, navigate roads, and more, often with greater accuracy than humans.
• Optimization Problems: In logistics, manufacturing, and energy management, AI optimizes resource allocation, production schedules, and energy consumption, enhancing efficiency and reducing costs.
• Creative Problem Solving: Beyond analytical tasks, AI also engages in creative problem-solving, generating art, music, and novel designs, showcasing its expanding role in creative industries.

These examples highlight AI’s broad problem-solving capabilities, showcasing its potential to transform industries and improve our understanding of complex systems.

4. Techniques AI Uses to Solve Problems

AI employs a variety of sophisticated techniques to address and solve problems, each tailored to the nature of the challenge at hand. These techniques not only highlight the versatility of AI but also its capacity for innovation and adaptation:

• Machine Learning (ML): ML algorithms allow AI to learn from data, identifying patterns and making predictions. This technique is particularly effective for problems where historical data can inform future decisions, such as predictive maintenance in manufacturing or recommendation systems in e-commerce.
• Deep Learning (DL) : A subset of ML, DL uses neural networks with many layers to process data in complex ways. It’s pivotal in image and speech recognition tasks, enabling functionalities like facial recognition systems and voice-activated assistants.

• Natural Language Processing (NLP): NLP allows AI to understand, interpret, and generate human language. This technique is at the core of chatbots, translation services, and sentiment analysis tools, solving the problem of machine interaction in human terms.
• Reinforcement Learning (RL): In RL, an AI system learns to make decisions by performing actions in an environment to achieve a goal. It’s used in robotics and gaming AI, where the machine learns optimal strategies through trial and error.
• Evolutionary Algorithms (EAs): EAs simulate the process of natural selection to generate high-quality solutions to optimization and search problems. They are useful in scheduling, logistics, and design optimization, where they can discover efficient solutions in complex search spaces.

5. Real-World Applications of AI Problem-Solving

The application of AI’s problem-solving capabilities is vast and varied, profoundly impacting various sectors:

• Healthcare: AI-driven diagnostic tools can analyze medical images to detect diseases early, and predictive models can forecast outbreaks or patient admissions, improving public health response and hospital management.
• Finance: AI enhances fraud detection, automates trading, and personalizes financial advice, making the financial sector more secure and tailored to individual needs.
• Transportation: Autonomous vehicles use AI to navigate safely, while logistics companies leverage AI for route optimization and supply chain management, significantly improving efficiency and reducing costs.
• Customer Service: AI-powered chatbots and virtual assistants provide 24/7 customer support, handling inquiries and solving problems with increasing sophistication.
• Environmental Conservation: AI assists in climate modeling, tracks wildlife populations, and optimizes renewable energy production, contributing to sustainable practices and conservation efforts.

These applications demonstrate AI’s transformative power in solving real-world problems, driving advancements across industries, and improving everyday life.

6. Challenges in AI’s Problem-Solving Capabilities

Despite its significant achievements, AI’s journey in problem-solving is not without challenges. These obstacles highlight the complexities of artificial intelligence and areas needing further development:

• Data Bias and Fairness: AI systems learn from data, which can contain biases reflecting historical inequalities or prejudices. Ensuring fairness and mitigating bias in AI’s decisions remains a significant challenge, requiring continuous efforts in data curation and algorithmic accountability.
• Explainability and Transparency: Many AI models, especially deep learning networks, are often described as “black boxes” due to their complex and opaque decision-making processes. Enhancing the explainability of AI systems is crucial for trust, ethical considerations, and regulatory compliance.
• Generalization Across Contexts: AI’s ability to generalize learned solutions to new, unseen problems varies greatly. Achieving a level of general intelligence, where AI can apply insights from one domain to another seamlessly, is a considerable challenge.
• Computational Resources and Energy Consumption: Advanced AI models demand significant computational power and energy, raising concerns about environmental impact and accessibility. Balancing performance with sustainability is an ongoing challenge.
• Security and Privacy: As AI becomes more integrated into critical systems, ensuring the security of AI models against adversarial attacks and protecting user privacy becomes paramount.

7. The Future of AI Problem-Solving

The future of AI problem-solving looks promising, with ongoing research and development poised to overcome current limitations and open new frontiers:

• Towards General AI: Efforts continue to develop more generalizable AI models that can perform a wide range of tasks with minimal specialized training, moving closer to the concept of General Artificial Intelligence (AGI).
• Quantum Computing and AI: The integration of quantum computing and AI holds the potential to revolutionize problem-solving capabilities, offering unprecedented computational power to tackle complex problems in optimization, cryptography, and more.

• Ethical AI Development: There is a growing focus on ethical AI development, emphasizing fairness, transparency, and accountability. This includes creating guidelines and frameworks to ensure AI’s positive impact on society.
• Collaborative AI: Future advancements may see more collaborative AI systems that work alongside humans, complementing human intelligence with AI’s computational efficiency in a hybrid approach to problem-solving.
• AI for Social Good: Increasingly, AI is being directed towards solving global challenges, including climate change, healthcare, and social inequality, highlighting its potential as a force for good.

8. References

• Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach . Pearson. A comprehensive textbook covering the fundamental concepts and techniques in AI.
• Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning . MIT Press. Provides an in-depth look at the methods and theories behind deep learning.
• RFC 8220 – Artificial Intelligence Markup Language : Discusses standards for AI-driven communication and data exchange protocols.
• RFC 9126 – Ethics in Artificial Intelligence : Proposes ethical considerations and guidelines for the development and deployment of AI systems.

Classroom Assessment Techniques, Part 3: Problem-solving, Application and Performance

By dr. diane chapman , executive director, ofd.

In earlier posts, I introduced the concept of using Classroom Assessment Techniques or CATs for comprehension and for synthesis, and critical and creative thinking . This post focuses on CATs for problem-solving, application and performance.

As a reminder, CATs were originally envisioned as simple, non-graded, in-class activities designed to give you and your students useful feedback on the teaching-learning process as it is happening.

Four CATs to Incorporate for Assessing Problem-solving, Application and Performance

These four CATs can be used effectively in both large and small courses and online and face-to-face. They are designed to focus on assessing student learning in critical and creative thinking and synthesis.

Student-generated Test Questions The purpose of the student-generated test questions is to give students the opportunity to evaluate the course topics, reflect on what they understand, and discover the aspects of effective test questions.

Process: Students are asked to write two test questions and model answers for specified topics, in a format consistent with course exams. You may want to give groups of students different topics or let them choose. You can then evaluate the questions and use the good ones as discussion prompts or to build an exam test bank. More about student generated test questions .

Directed Paraphrasing The purpose of directed paraphrasing is to analyze the students’ ability to comprehend materials and transfer knowledge. Process: Students are asked to write in layperson’s terms, about something they learned. The instructor or students should choose a target audience such as a grandparent, parent, employer, sibling, etc. Students can either turn these in or share with each other and discuss their level of success in translating the material. Examples of directed paraphrasing .

Application Cards The purpose of application cards is to quickly determine whether or not students understand the practical applications of what they have learned. Students are forced to link new information with prior knowledge.

Process: After students have been introduced to some principle, generalization, theory, or procedure, you would pass out index cards and ask students to write down at least one possible, real-world application for what they have just learned. To assess, the responses can quickly be separated into four groups — great, acceptable, marginal, and not acceptable. Responses should be discussed in the next class. More about application cards .

Documented Problem Solutions The purposes of document problem solutions is to assess how students solve problems and to evaluate how well students comprehend and can describe their problem-solving methods.

Process: Students are asked to document the specific steps they take in attempting to solve a problem. The instructor selects two or three problems students have studied in recent weeks. The instructor solves the problems, noting how long it took to solve it and the steps each solution required. Problems should be able to be solved in less than thirty minutes. You should distribute the problem and directions to students making clear that it is not a test or a quiz. It is more important for students to explain how they tried to solve the problems than to get the right answers. Example of documented problem solutions .

What About Online Courses?

All of these CATs can be done in an online course or face-to-face and completed in-class or be alternatively done as homework assignments.

Formative Assessment

Remember that CATs are primarily for formative assessment – to see where your students are while you still have time to change, adapt or add to instruction. Because they are not intended to be graded, most are suitable for both small and large courses. Designing these types of assessments into your courses on a regular basis can also reduce student anxiety over graded assessments by helping them understand your expectations. If you need help in designing a CAT for your course, contact us at the Office of Faculty Development. We also want to hear about your stories of implementing CATs into your courses.

• Angelo, T. A., & Cross, K. P. (1993). Classroom assessment techniques: A handbook for college teachers (2nd Ed.). San Francisco: Jossey-Bass.
• Center for Enhancement of Teaching and Learning (n.d.). 50 Classroom Assessment Techniques (CATs) Available at https://www.uky.edu/celt/50-classroom-assessment-techniques-cats

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Do You Understand the Problem You’re Trying to Solve?

To solve tough problems at work, first ask these questions.

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

• Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
• Find more episodes of HBR IdeaCast
• Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

  In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.  

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

We’ll be back next Wednesday with another hand-picked conversation about leadership from the Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

We’re a production of Harvard Business Review. If you want more podcasts, articles, case studies, books, and videos like this, find it all at HBR dot org.

This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Maureen Hoch, Adi Ignatius, Karen Player, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener.

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You can master Computer Vision, Deep Learning, and OpenCV - PyImageSearch

Algorithms Artificial Intelligence Data Science Deep Learning Machine Learning

Exploring the Landscape of Machine Learning: Techniques, Applications, and Insights

by Hector Martinez on April 1, 2024

Table of Contents

Introduction: the power of machine learning in modern industries, what is machine learning, understanding the core types of machine learning techniques, supervised learning: from basics to real-world applications explained, unsupervised learning explained: discovering hidden patterns, bridging the gap with semi-supervised learning: enhancing data understanding, core machine learning techniques for business innovation, deep learning: unleashing the power of neural networks, leveraging transfer learning for efficient ai development, federated learning: privacy-preserving machine learning, meta-learning: teaching ai to learn more effectively, deep learning breakthroughs, generative adversarial networks (gans): innovations in synthetic data, transformers in nlp: beyond conventional models, reinforced learning: strategies for a model to learn from interaction, machine learning for solving real-world problems, understanding different machine learning problem types, solving classification problems with machine learning, solving regression problems through machine learning techniques, clustering problems: unsupervised learning approaches, detecting anomalies: unsupervised learning for anomaly detection, optimizing decision-making with reinforcement learning: strategies and applications, leveraging machine learning for strategic advantages across industries, exploring the backbone of ai: a guide to machine learning algorithms, comprehensive guide to machine learning algorithms, decision trees: key to classification and regression, random forests for enhanced prediction accuracy, support vector machines (svm) in machine learning.

• Neural Networks: The Brain Behind AI’s Decision-Making

K-Nearest Neighbors (KNN): A Go-To Algorithm for Precision

Principal component analysis (pca): simplifying data with dimensionality reduction, clustering algorithms: grouping data with machine learning, the critical role of labels in machine learning algorithms, harnessing semi-supervised learning to reduce labeling costs, exploring unsupervised learning: beyond labels, maximizing rewards with reinforcement learning, leveraging analytical learning for data-driven decisions, high-dimensional data with analytical models, summary: mastering machine learning for real-world solutions.

The field of machine learning is taking the world by storm, revolutionizing various industries that range from healthcare to finance to transportation. With the massive amounts of data that businesses and organizations now generate, machine learning algorithms have become a critical tool for extracting insights and making informed decisions. There are different types of machine learning available, each with its own unique advantages and drawbacks. In this article, we’ll delve into the four primary forms of machine learning: supervised learning, unsupervised learning, semi-supervised learning, and reinforced learning.

Note: This blog post is meant to be a guide to the ever-changing landscape of AI and Machine Learning. If you already have some familiarity with fundamental topics, you probably don’t need this, and can check out some of our more advanced blog posts here .

Machine learning (ML) is a type of artificial intelligence (AI) that’s focused on creating algorithms that can learn from data and improve their performance over time. Instead of explicitly programming them for every task, machine learning algorithms are designed to automatically identify patterns in data and use those patterns to make predictions or decisions.

To get a more grounded, code-first introduction to machine learning, read here .

This is the most common type of machine learning, and it is used when the data is labeled. In this case, the algorithm learns to map inputs to outputs based on examples of labeled data. The input data is referred to as features, and the output data is referred to as the label or target. Supervised learning aims to use these labeled examples to train the algorithm to make accurate predictions on new, unlabeled data.

There are two main types of supervised learning: classification and regression. Classification is used when the output is a categorical variable, and the algorithm needs to predict the category where the input data belongs. Examples of classification tasks include image recognition, sentiment analysis, and spam detection. Regression is used when the output is a continuous variable, and the algorithm needs to predict a numerical value. Examples of regression tasks include predicting housing prices, weather forecasting, and stock market analysis.

Unsupervised learning is used when the data is unlabeled. In this case, the algorithm learns to find patterns and relationships in the data without explicit guidance. Unsupervised learning aims to explore the data structure and discover any hidden patterns or groupings.

Several types of unsupervised learning include clustering, dimensionality reduction, and anomaly detection. Clustering is used to group similar data points based on their similarities, while dimensionality reduction is used to reduce the number of features in the data to simplify the problem. Finally, anomaly detection is used to identify unusual data points that do not fit the normal patterns of the data.

Semi-supervised learning is used when the data is partially labeled. In this case, the algorithm uses labeled and unlabeled data to make predictions. Semi-supervised learning aims to use the labeled data to guide the learning process and improve the accuracy of the predictions.

Semi-supervised learning is often used when data labeling is expensive or time-consuming, such as in medical imaging or natural language processing. By using the available labeled data to guide the learning process, semi-supervised learning can achieve high levels of accuracy with less labeled data than would be required for supervised learning.

Machine learning is a powerful tool that can help businesses and organizations make better decisions and gain new insights into their data. Understanding the different types of machine learning is essential for choosing the right approach for a given problem. Whether you are working with labeled or unlabeled data, or whether you need to learn through trial and error, there is a type of machine learning that can help you achieve your goals.

The field of machine learning is advancing rapidly, and new techniques and algorithms are being developed at an ever-increasing rate. In this blog post, we will explore some of the latest and cutting-edge machine learning techniques that are currently making waves in the industry.

Some of our tutorials provide you with the tools and techniques required for business innovation in the field of Deep Learning and Computer Vision.

1. Deep Learning

• Self-Driving Cars: Deep learning algorithms excel at object detection and recognition, which is crucial for self-driving cars to navigate safely. They can identify pedestrians, vehicles, traffic signs, and more in real-time.
• Medical Diagnosis: Deep learning can analyze medical images like X-rays, mammograms, and MRIs to detect abnormalities or diseases, aiding doctors in diagnosis and treatment planning.
• Facial Recognition: Deep learning powers facial recognition systems used for security purposes, access control, and even personalized marketing.

2. Embedded Systems

• Internet of Things (IoT): Embedded systems equipped with computer vision capabilities can be used in smart homes for tasks like object recognition (security cameras) or facial recognition (smart door locks).
• Industrial Automation: Embedded systems with machine learning can perform real-time quality control in factories, identify defects in products, or predict equipment maintenance needs.
• Robotics: Embedded systems with computer vision allow robots to navigate their environment, identify objects for manipulation, and interact with the physical world more intelligently.

3. Optical Character Recognition (OCR)

• Document Automation: OCR can automate data entry tasks by extracting text from scanned documents, invoices, or receipts, saving time and reducing errors.
• Self-Service Systems: Libraries and banks use OCR scanners to automate book check-in/out or process checks for deposit.
• Accessibility Tools: OCR technology can convert printed text into audio for visually impaired individuals, making documents and information more accessible.

4. Machine Learning

• Recommendation Systems: Machine-learning algorithms power recommendation systems on e-commerce platforms or streaming services, suggesting products or content users might be interested in.
• Fraud Detection: Machine learning can analyze financial transactions to identify fraudulent activity in real-time, protecting users from financial harm.
• Spam Filtering: Machine learning algorithms can analyze email content to identify and filter spam messages, keeping your inbox clean and organized.

These are just a few examples, and the potential applications of these technologies continue to grow as computer vision and machine learning advancements accelerate. Explore these resources on PyImageSearch to delve deeper into the practical implementations of these techniques in various real-world scenarios.

Deep learning is a subset of machine learning based on artificial neural networks. It has been a hot topic in the machine-learning community for several years. It has been used in a wide range of applications, from speech recognition to image classification to natural language processing.

The key advantage of deep learning is its ability to learn and extract features from large, complex datasets. This is achieved by building a hierarchy of neural networks, where each layer extracts increasingly complex features from the input data. Deep learning has also been shown to outperform traditional machine learning algorithms in many tasks.

Transfer learning is a technique that allows a pre-trained model to be used for a new task with minimal additional training. This is achieved by leveraging the knowledge that the pre-trained model has already learned and transferring it to the new task. Read more about the practical aspects of Transfer Learning in the tutorial from Figure 1 .

Transfer learning has become popular in recent years because it can significantly reduce the data and training time required for a new task. It has been used in a wide range of applications, including natural language processing, image recognition, and speech recognition. Here’s how it can be applied to various applications:

1. Object Detection

• Pre-trained Models: Popular choices include VGG16, ResNet50, or InceptionV3 trained on ImageNet (a massive image dataset with thousands of object categories).
• Freeze the initial layers of the pre-trained model (these layers learn generic features like edges and shapes).
• Add new layers on top specifically designed for object detection (like bounding box prediction).
• Train the new layers with your custom object detection dataset.
• Benefits: Significantly reduces training time compared to training from scratch and leverages pre-learned features for better object detection accuracy.

2. OCR (Optical Character Recognition)

• Pre-trained Models: These are trained on large text datasets like MNIST (handwritten digits) or COCO-Text (images with text captions).
• Freeze the initial layers responsible for extracting low-level image features.
• Add new layers (e.g., convolutional layers) specifically designed for character recognition.
• Train the new layers with your custom dataset, which contains images of the specific text format you want to recognize (e.g., invoices, receipts, license plates).
• Benefits: Faster training and improved accuracy for recognizing specific text formats compared to training from scratch.

3. Image Classification

• Pre-trained Models: Similar to object detection, models like VGG16 or ResNet50 can be used.
• Freeze the initial layers of the pre-trained model.
• Add a new fully connected layer at the end with the number of neurons matching your classification categories.
• Train the new layer with your custom image dataset labeled for your specific classification task (e.g., classifying types of flowers and different breeds of dogs).
• Benefits: Reduces training time and leverages pre-learned features for improved image classification accuracy on new datasets.

Additional Points:

• Fine-tuning the Model: To achieve optimal results, you can adjust the learning rate of the newly added layers compared to the frozen pre-trained layers.
• Transfer Learning Limitations: While powerful, transfer learning might not be ideal for entirely new visual concepts not present in the pre-trained model’s training data. In such cases, custom model training from scratch might be necessary.

By leveraging transfer learning, we can achieve significant performance improvements in various computer vision tasks with less training data and computational resources.

Federated learning is a technique that allows multiple devices to collaboratively learn a model without sharing their data. This is achieved by training the model locally on each device and then aggregating the results to create a global model.

Federated learning has become popular in applications where data privacy is a concern, such as healthcare and finance. It allows models to be trained on data that cannot be centralized, such as data stored on individual devices or in different geographic locations.

Meta-learning is a technique that allows a model to learn how to learn. This is achieved by training the model on a variety of tasks and environments so it can quickly adapt to new tasks and environments.

Meta-learning has been used in a wide range of applications, from computer vision to natural language processing. It has the potential to significantly reduce the amount of training data and time required for a new task, making it a powerful tool for machine learning.

These are just a few of the many new and cutting-edge machine-learning techniques being developed. As the field of machine learning continues to advance, we can expect to see many more exciting developments in the coming years. By staying up-to-date with the latest trends and techniques, you can stay ahead of the curve and unlock the full potential of machine learning in your organization.

Generative Adversarial Networks (GANs) are a type of deep learning model that has gained a lot of attention in recent years. GANs consist of two neural networks: a generator and a discriminator. The generator creates synthetic data that is similar to the real data, and the discriminator tries to distinguish between the real and synthetic data.

The goal of GANs is to train the generator to create synthetic data that is indistinguishable from real data. This data can be used for tasks such as image synthesis and data augmentation. GANs have also been used in other applications, such as generating realistic 3D models and creating deepfakes.

Transformers are a type of deep learning model that has gained a lot of attention in recent years, particularly in the field of natural language processing (NLP). The transformer architecture was introduced by Vaswani et al. (2017) and has since become a popular choice for a wide range of NLP tasks.

Traditional NLP models, such as recurrent neural networks (RNNs) and convolutional neural networks (CNNs), process input sequences in a linear fashion. This can lead to difficulties in modeling long-range dependencies and capturing relationships between words that are far apart in the input sequence. Transformers, on the other hand, transformers use a self-attention mechanism to process input sequences in a parallel fashion, allowing them to model long-range dependencies more effectively.

In a transformer, the input sequence is first embedded into a high-dimensional vector space. Then, multiple layers of self-attention and feedforward neural networks are applied to the input sequence in parallel. The self-attention mechanism allows the model to focus on different parts of the input sequence and learn to associate words that are far apart in the sequence. The feedforward neural networks will enable the model to learn more complex interactions between the words.

At PyImageSearch, we have crafted a three-part tutorial on Transformers (shown in Figure 2 ) to take you from the basics of attention mechanism to creating your own transformer for Neural Machine Translation.

One of the transformers’ key advantages is their ability to handle variable-length input sequences. This is particularly useful in NLP, where input sequences can vary greatly in length. In addition, transformers have been shown to outperform traditional NLP models on a wide range of tasks, including language modeling, machine translation, and text classification.

One of the most popular implementations of transformers is the BERT (Bidirectional Encoder Representations from Transformers) model, which was introduced by Google in 2018. BERT uses a transformer-based architecture to generate contextualized word embeddings, which are then used as input to downstream NLP tasks. BERT has achieved state-of-the-art performance on many NLP tasks, including sentiment analysis, question answering, and named entity recognition.

Another popular implementation of transformers is the GPT (Generative Pre-training Transformer) model, which was introduced by OpenAI in 2018. GPT uses a transformer-based architecture to generate text, and it has been used to generate realistic, human-like text in a wide range of applications, from chatbots to creative writing.

Transformers are a powerful type of deep learning model that has revolutionized the field of NLP. Their ability to handle variable-length input sequences and model long-range dependencies has made them a popular choice for a wide range of NLP tasks. As the field of NLP continues to advance, we can expect to see many more exciting developments in the area of transformer-based models.

Reinforced learning is used when the algorithm needs to learn through trial and error. In this case, the algorithm interacts with an environment and receives rewards or penalties for its actions. Reinforced learning aims to learn the optimal policy, or set of actions, that maximizes the cumulative reward over time.

Reinforced learning is often used in robotics, gaming, and autonomous vehicles. In these cases, the algorithm must learn how to navigate a complex environment and make decisions that lead to the desired outcome. By receiving feedback in rewards or penalties, the algorithm can learn from its mistakes and improve over time.

Machine learning is a powerful tool for solving a wide range of problems in many different industries. By analyzing large datasets and extracting patterns and insights, machine learning algorithms can help businesses and organizations make better decisions, improve efficiency, and reduce costs. In this blog post, we will explore some of the types of problems that can be solved with machine learning.

Classification problems are one of the most common types of problems that can be solved with machine learning. In a classification problem, the goal is to assign a label to an input based on its features. For example, a machine learning algorithm could be used to classify emails as spam or not spam or to classify images as dogs or cats.

Classification problems are often solved using supervised learning algorithms, such as decision trees, support vector machines, and neural networks. These algorithms learn to map input features to output labels by analyzing examples of labeled data.

Regression problems are another common type of problem that can be solved with machine learning. In a regression problem, the goal is to predict a continuous output value based on the input features. For example, a machine learning algorithm could be used to predict housing prices based on features such as square footage, number of bedrooms, and location.

Regression problems are also often solved using supervised learning algorithms, such as linear regression, decision trees, and neural networks. These algorithms learn to map input features to output values by analyzing examples of labeled data.

Clustering problems are a type of unsupervised learning problem. In a clustering problem, the goal is to group similar items based on their features. For example, a machine learning algorithm could be used to cluster customers based on their purchasing habits, or to group documents based on their content.

Clustering problems are often solved using unsupervised learning algorithms, such as k-means clustering, hierarchical clustering, and density-based clustering. These algorithms learn to identify patterns in the data by analyzing examples of unlabeled data.

Anomaly detection problems are another type of unsupervised learning problem. In an anomaly detection problem, the goal is to identify unusual data points that do not fit the normal patterns of the data. For example, a machine learning algorithm could be used to detect fraudulent credit card transactions based on patterns in the transaction data.

Anomaly detection problems are often solved using unsupervised learning algorithms, such as density-based clustering and autoencoders. These algorithms learn to identify patterns in the data by analyzing examples of unlabeled data.

Reinforcement learning problems are a type of machine learning problem where the goal is to learn a policy, or set of actions, that maximizes a reward signal over time. For example, a machine learning algorithm could be used to learn to play a game or navigate a robot through a maze.

Reinforcement learning problems are often solved using reinforcement learning algorithms, such as Q-learning and policy gradient methods. These algorithms learn to optimize a policy by exploring the environment and receiving feedback in the form of rewards or penalties.

Machine learning can solve a wide range of problems in many different industries. By using machine learning algorithms to analyze large datasets, businesses, and organizations can gain new insights and make better decisions, leading to improved efficiency and reduced costs.

Machine learning algorithms are the backbone of many artificial intelligence (AI) applications. Several types of algorithms are commonly used in machine learning, each with its own strengths and weaknesses. In this blog post, we will explore some of the different types of algorithms in machine learning.

Decision trees are a type of supervised learning algorithm that is commonly used for classification and regression tasks. The algorithm works by recursively splitting the data based on the values of the input features until each leaf node contains a single output value. Decision trees are easy to interpret and can handle both categorical and continuous data.

Random forests are a type of ensemble learning algorithm that combines multiple decision trees to improve the accuracy of the predictions. The algorithm works by creating a set of decision trees, each trained on a random subset of the data and features. Random forests are often used for classification and regression tasks and can handle large datasets with high-dimensional features.

Support vector machines (SVMs) are a type of supervised learning algorithm that is commonly used for classification and regression tasks. The algorithm works by finding a hyperplane that maximally separates the data into different classes or predicts a continuous output value. SVMs can handle both linear and nonlinear data and are effective for high-dimensional data with a small number of training examples.

Neural Networks: The Brain Behind AI’s Decision-Making

Neural networks are a type of supervised learning algorithm that is commonly used for classification and regression tasks. The algorithm works by simulating the function of the human brain with a network of interconnected nodes that process the input data. Neural networks are effective for high-dimensional data with complex relationships between the input features.

K-nearest neighbors (KNN) is a type of supervised learning algorithm that is commonly used for classification and regression tasks. The algorithm works by finding the k nearest neighbors to a given data point and using their labels or values to predict the output for the new data point. KNN can handle both continuous and categorical data and is effective for small datasets with low-dimensional features.

Principal component analysis is an unsupervised learning algorithm that is commonly used for dimensionality reduction. The algorithm works by finding the principal components of the data, which are the linear combinations of the input features that capture the most variance in the data. PCA can be used to reduce the dimensionality of the data, making it easier to visualize and analyze.

Clustering algorithms are unsupervised learning algorithms that group similar data points based on their features. There are several types of clustering algorithms, including k-means, hierarchical clustering, and density-based clustering. Clustering algorithms can be used to identify patterns in the data and find hidden structures.

As you can see, many different types of algorithms are used in machine learning. Each algorithm has its own strengths and weaknesses, and the choice of algorithm depends on the type of data and the specific task at hand. By using the right algorithm for the job, businesses and organizations can gain new insights and make better decisions based on the analysis of their data.

Labels are an essential component of many machine-learning algorithms. In supervised learning, labels are used to train a model to predict output values based on input features. The process of labeling data is time-consuming and requires expertise, but it is a necessary step in building effective machine-learning models.

In supervised learning, labels are attached to each data point in the training set, indicating the correct output value for that data point. For example, if the input is an image, the label might indicate whether the image contains a dog or a cat. If the input is a sentence, the label might indicate the sentiment of the sentence (positive, negative, or neutral).

Labeling data is typically done manually, either by humans or by using other machine learning algorithms. Human labeling can be time-consuming and expensive, especially for large datasets. However, it is often necessary to ensure high-quality labels, particularly for complex tasks or tasks that require human expertise.

One way to reduce the cost and time required for labeling is through semi-supervised learning. In semi-supervised learning, a small portion of the data is labeled, and the rest of the data is left unlabeled. The model is then trained on the labeled data, and the knowledge gained from this training is used to make predictions for the unlabeled data. This can be a cost-effective way to train a machine learning model, particularly for large datasets.

In addition to supervised learning, labels are also used in unsupervised learning algorithms. In clustering algorithms, for example, the goal is to group similar data points based on their features. While the data points may not have explicit labels, the clusters themselves can be used to infer labels or insights about the data.

Labels are also used in reinforcement learning, where the goal is to learn a policy that maximizes a reward signal over time. In this case, the reward signal acts as a label, indicating the correct action to take in a given situation.

You probably noticed by now that labels are an essential component of many machine learning algorithms. While the process of labeling data can be time-consuming and expensive, it is necessary to train effective machine learning models. By using labeled data, businesses and organizations can gain new insights and make better decisions based on the analysis of their data.

Analytical learning is a type of machine learning that involves using mathematical models and statistical analysis to make predictions or decisions based on data. It is one of the most common approaches to machine learning and is used in a wide range of applications, from business analytics to healthcare to autonomous vehicles.

Analytical learning is often used in supervised learning, where the goal is to predict output values based on input features. In analytical learning, a model is trained on a set of labeled data using statistical methods and mathematical models. The model then uses this knowledge to make predictions on new, unseen data.

Tabular data remains a significant and crucial format. Here are some reasons why:

• Structured and Organized: Tabular data is inherently organized in rows and columns, making it easy for humans and computers to understand and analyze.
• Legacy Systems: Many businesses and organizations still rely on databases and spreadsheets that store information in a tabular format.
• Analysis Foundation: Tabular data serves as the foundation for many machine learning algorithms, making it a vital tool for extracting insights.

Several types of analytical learning models are commonly used in machine learning. These include linear regression, logistic regression, decision trees, random forests, support vector machines (SVMs), and artificial neural networks (ANNs). Each type of model has its own strengths and weaknesses, and the choice of model depends on the specific problem and the characteristics of the data. In reality, a variety of neural network architectures can be employed to understand heterogeneous tabular data, as shown in Figure 3 .

In addition to supervised learning, analytical learning can also be used in unsupervised learning, where the goal is to identify patterns and relationships in the data. In unsupervised learning, the model is not given explicit output labels, but instead, it is used to group or cluster similar data points based on their features. Common unsupervised learning algorithms include k-means clustering, hierarchical clustering, and principal component analysis (PCA).

One key advantage of analytical learning is its ability to handle large datasets and complex relationships between the input features. By using statistical methods and mathematical models, analytical learning can extract patterns and insights from the data that may not be obvious to humans.

However, analytical learning also has limitations. For example, it may need help to handle high-dimensional data with many input features and be sensitive to outliers and noise in the data. In addition, analytical learning may not be suitable for tasks that require human expertise or judgment.

Analytical learning is a powerful tool in machine learning that can be used to make predictions or decisions based on data. It is a widely used approach that involves using mathematical models and statistical analysis to extract patterns and insights from the data. By using analytical learning, businesses and organizations can gain new insights and make better decisions based on the analysis of their data.

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In this post, we learned about various types of machine learning, such as supervised, unsupervised, and reinforcement learning, as well as insights into deep learning and its applications (e.g., GANs and transfer learning). Additionally, we also introduced different problem types, including classification, regression, clustering, and anomaly detection, and explored algorithms like decision trees, random forests, and neural networks. Now, you’re ready to dive deeper and start training your own machine-learning models to solve interesting problems. Be sure to check out our other blogs or, even better, join PyImageSearch University , where you’ll get videos, code downloads, and all the help you need to be successful in machine learning.

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How To Debug Django: Practical Steps For Efficient Problem Solving

Facing bugs in Django can be daunting, but with the right techniques, debugging becomes a breeze. This article sheds light on practical methods and tools to help you tackle issues head-on, ensuring smoother development experiences and more robust applications.

Debugging is a crucial skill for every programmer, and when working with Django, it becomes especially important. This article offers practical insights and techniques to help you identify and fix issues effectively. With a focus on real-world scenarios, we aim to enhance your problem-solving skills in the Django framework.

💡 How To Debug Django

• Efficient Django debugging relies on tools like the debugger, error messages, and logs for issue resolution.
• Emphasizing error handling strategies and debugging highlights the importance of structured approaches in maintaining web app robustness.
• Django's Debug Toolbar simplifies debugging by providing insights into performance metrics like database queries and template rendering.
• This guide simplifies Django debugging with insights into tools, strategies, and the Debug Toolbar for effective troubleshooting and a seamless web app experience.

Setting Up The Debugging Environment

Identifying common django errors, using django debugging tools, debugging django templates, debugging django models, debugging django views, debugging django forms, logging and monitoring in django, optimizing query performance, frequently asked questions.

Setting up the debugging environment is a foundational step in addressing Django-related issues. Debugging environment and Django settings are pivotal in this process. Before diving into debugging, ensure your environment is correctly configured to facilitate the identification and resolution of bugs.

Configuring Django Settings

Installing debug toolbar, middleware configuration, internal ips setting.

Start by adjusting the DEBUG setting in your Django project’s settings file. This setting controls whether your application is in debugging mode. When DEBUG is set to True, Django will display detailed error pages when something goes wrong.

Another useful tool is the Django Debug Toolbar . It provides insights into the performance and configuration of your Django application. To install it, use the following command:

Once installed, add the Debug Toolbar’s middleware to the MIDDLEWARE setting in your settings file. Ensure it’s at the top of the list to capture all requests and responses.

Lastly, set INTERNAL_IPS in your settings file to include the IP address of your development machine. This step is necessary for the Debug Toolbar to be displayed.

Identifying common Django errors is a key skill that can save developers a lot of time and frustration. Error messages and tracebacks are your guides in this journey, pointing you towards the source of the problem.

Django’s Error Messages

Understanding tracebacks, common error types, debugging 404 errors.

Django provides informative error messages to help you understand what’s gone wrong. For instance, a FieldError indicates that a model field name is incorrect.

Tracebacks are another valuable tool. They provide a stack trace of your code, showing the path the interpreter took leading up to the error. Analyzing tracebacks helps in locating the erroneous part of your code.

Some of the common Django error types you might encounter include:

• ImproperlyConfigured : This error occurs when Django detects a problem with your project’s configuration.
• ValidationError : Django raises this error when an object doesn’t meet the validation requirements.
• DoesNotExist : This error is raised by the ORM when a requested object can’t be found.

404 errors are common and occur when the requested URL doesn’t match any of your URL patterns. Django’s DEBUG mode provides a helpful error page to debug this issue, showing which URL patterns were tried and why they didn’t match.

Utilizing Django debugging tools effectively is essential for diagnosing and resolving issues in your application. Django Debug Toolbar and logging are two powerful tools that can significantly aid in this process.

Django Debug Toolbar

Template panel.

The Django Debug Toolbar is a configurable set of panels displaying various debug information about the current request/response. It assists in understanding SQL queries, template usage, cache utilization, and more.

One of the most useful panels is the SQL Panel , which displays all SQL queries executed during the processing of the request. It helps in identifying redundant queries and optimizing database access.

The Template Panel is another valuable resource, showing all templates that were used and the context data that was passed to them. This panel is particularly helpful for debugging template rendering issues.

In addition to the Debug Toolbar, Django’s built-in logging framework is a versatile tool for recording debug information, errors, and warnings. Configuring logging appropriately is crucial for effective debugging.

Debugging Django templates can be challenging due to their unique syntax and behavior. However, with a strategic approach and the right tools, you can efficiently identify and resolve template issues. Template errors and context data are crucial aspects to focus on.

Identifying Template Errors

Examining context data, using the debug toolbar, template filters and tags.

Django provides error messages specifically for templates, making it easier to pinpoint issues. For instance, if you use an undefined variable, Django will raise a TemplateDoesNotExist error.

Examining the context data passed to a template is essential for debugging. You can print the context data directly in the template to inspect the available variables.

The Django Debug Toolbar offers a Template Panel that displays all templates used during the rendering of a page, along with their context data. This feature is invaluable for debugging template issues.

Understanding the behavior of template filters and tags is vital. If they are not used correctly, they can lead to unexpected results. Testing each filter and tag individually can help isolate the problem.

Debugging Django models is a fundamental aspect of ensuring the integrity and functionality of your applications. Model fields and methods are central elements that require careful attention during the debugging process.

Inspecting Model Fields

Debugging model methods, using django shell, examining database queries.

Incorrectly configured model fields can lead to various issues. It’s essential to verify the field types , options , and constraints to ensure they align with your data requirements.

Model methods can contain logic errors or misuse of the Django ORM. Debugging these methods involves checking the querysets and operations performed on the model instances.

The Django shell is a powerful tool for interacting with your models. You can use it to create, retrieve, update, and delete model instances, allowing you to test and debug model behavior interactively.

Examining the SQL queries generated by the Django ORM can provide insights into model-related issues. The Django Debug Toolbar’s SQL Panel is particularly helpful for this purpose.

Debugging Django views is a critical step in ensuring that your application responds appropriately to user requests. View functions and class-based views are the main components that need thorough examination during debugging.

Analyzing View Functions

Debugging class-based views, utilizing logging, testing with client.

View functions handle HTTP requests and return HTTP responses. Debugging involves checking the logic , input validation , and context data within these functions.

Class-based views offer a structured way to handle HTTP requests. Debugging them involves examining the methods and attributes of the view class.

Logging is an invaluable tool for debugging views. By adding log statements in your views, you can capture information about the execution flow and data being processed.

Django’s test Client allows you to simulate HTTP requests in your tests, making it a valuable tool for debugging views. You can use it to test the response status, content, and context data.

Debugging Django forms is essential for ensuring user data is correctly validated and processed. Form fields , validation methods , and error messages are key areas to focus on during the debugging process.

Examining Form Fields

Debugging validation methods, reviewing error messages, testing form submission.

Form fields are the building blocks of Django forms. Ensuring that each field has the correct field type , validators , and options is crucial for form functionality.

Custom validation methods in forms should be thoroughly debugged. These methods are responsible for enforcing custom validation rules and raising validation errors.

Understanding and reviewing the error messages generated by form validation is essential. These messages guide the user in providing valid input and need to be clear and accurate.

Simulating and testing form submission in different scenarios help identify issues in form processing. Django’s test Client can be used to simulate POST requests with form data.

Logging and monitoring are indispensable practices in Django development, aiding in tracking application behavior, identifying issues, and optimizing performance. Logging configuration and third-party monitoring tools are essential components in this process.

Configuring Django Logging

Implementing logging in code, utilizing monitoring tools, analyzing log data.

Django comes with a built-in logging framework. Configuring logging settings in your settings.py file allows you to specify log levels, handlers, and formats.

Once configured, you can implement logging throughout your code using loggers . Loggers allow you to capture messages at different severity levels.

In addition to logging, integrating third-party monitoring tools like Sentry can provide real-time error tracking and performance insights.

Regularly analyzing your log data can help identify patterns, detect anomalies, and optimize application performance. Tools like ELK Stack can assist in aggregating and visualizing log data.

Optimizing query performance is pivotal in enhancing the efficiency and speed of a Django application. Database indexing , query optimization , and caching are fundamental techniques to achieve this.

Leveraging Database Indexing

Utilizing select related, applying prefetch related, implementing caching, analyzing query performance.

Database indexing significantly speeds up data retrieval. Ensure that fields used in filtering and ordering are properly indexed.

The select_related method is used for single-valued relationships , reducing the number of queries by performing a SQL join and including related object data in the original query.

For multi-valued relationships , prefetch_related performs separate lookups for each relationship and does the “joining” in Python, reducing the overall number of queries.

Caching is a powerful technique to store and serve previously computed results, thereby reducing database hits and improving performance.

Regularly analyzing and optimizing your queries using Django’s database API and tools like Django Debug Toolbar can provide insights and improvements.

What Are the Best Practices for Debugging Django Applications?

Utilize Django’s built-in debugging tools, leverage logging and monitoring, optimize query performance, and familiarize yourself with common Django errors. Regularly analyze log data and use third-party tools like Django Debug Toolbar for additional insights.

How Can I Optimize the Performance of Database Queries in Django?

Leverage database indexing for frequently queried fields, utilize select_related and prefetch_related for relationship fields, implement caching strategies, and regularly analyze and optimize your queries using Django’s database API and additional tools.

Are There Any Third-Party Tools Recommended for Debugging Django?

Yes, tools like Django Debug Toolbar, Sentry, and ELK Stack are highly recommended for debugging, monitoring, and log analysis in Django applications.

How Can I Identify and Handle Common Errors in Django?

Familiarize yourself with Django’s error reporting and exception handling. Utilize logging to capture error messages and stack traces. Regularly review error logs and address common issues like template errors, model errors, and form validation errors.

What Is the Role of Caching in Django, and How Can It Be Implemented?

Caching stores and serves previously computed results, reducing database hits and improving performance. Django provides a caching framework that allows you to cache data using various backends, such as memory, file system, or databases.

Let’s test your knowledge!

In Django, which method can be used to optimize the performance of database queries by reducing the number of SQL queries for single-valued relationships?

Continue learning with these django guides.

• How To Update Django: Steps For A Successful Upgrade
• How To Install Django In Ubuntu: Step-By-Step Instructions For A Smooth Setup
• How To Reset Django Admin Password
• How To Start Django Server: Straightforward Steps For Quick Setup
• How To Install Django Step By Step

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3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Learning objectives.

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

• Solve applications using properties of triangles

Use the Pythagorean Theorem

• Solve applications using rectangle properties

Be Prepared 3.12

Before you get started, take this readiness quiz.

Simplify: 1 2 ( 6 h ) . 1 2 ( 6 h ) . If you missed this problem, review Example 1.122 .

Be Prepared 3.13

The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle. If you missed this problem, review Example 1.26 .

Be Prepared 3.14

Solve: A = 1 2 b h A = 1 2 b h for b when A = 260 A = 260 and h = 52 . h = 52 . If you missed this problem, review Example 2.61 .

Be Prepared 3.15

Simplify: 144 . 144 . If you missed this problem, review Example 1.111 .

Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

Solve Geometry Applications.

• Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
• Step 2. Identify what we are looking for.
• Step 3. Label what we are looking for by choosing a variable to represent it.
• Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
• Step 7. Answer the question with a complete sentence.

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word vertex is vertices . All triangles have three vertices . Triangles are named by their vertices: The triangle in Figure 3.4 is called △ A B C . △ A B C .

The three angles of a triangle are related in a special way. The sum of their measures is 180 ° . 180 ° . Note that we read m ∠ A m ∠ A as “the measure of angle A.” So in △ A B C △ A B C in Figure 3.4 ,

Because the perimeter of a figure is the length of its boundary, the perimeter of △ A B C △ A B C is the sum of the lengths of its three sides.

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a 90 ° 90 ° angle with the base. We will draw △ A B C △ A B C again, and now show the height, h . See Figure 3.5 .

Triangle Properties

For △ A B C △ A B C

Angle measures:

• The sum of the measures of the angles of a triangle is 180 ° . 180 ° .
• The perimeter is the sum of the lengths of the sides of the triangle.
• The area of a triangle is one-half the base times the height.

Example 3.34

The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.

Try It 3.67

The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle.

Try It 3.68

The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle.

Example 3.35

The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side?

Try It 3.69

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Try It 3.70

The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side?

Example 3.36

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

Try It 3.71

The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

Try It 3.72

A triangular tent door has an area of 15 square feet. The height is five feet. What is the base?

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one 90 ° 90 ° angle, which we usually mark with a small square in the corner.

Right Triangle

A right triangle has one 90 ° 90 ° angle, which is often marked with a square at the vertex.

Example 3.37

One angle of a right triangle measures 28 ° . 28 ° . What is the measure of the third angle?

Try It 3.73

One angle of a right triangle measures 56 ° . 56 ° . What is the measure of the other small angle?

Try It 3.74

One angle of a right triangle measures 45 ° . 45 ° . What is the measure of the other small angle?

In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 3.38

The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.

Try It 3.75

The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.

Try It 3.76

The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem . This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a 90 ° 90 ° angle, marked with a small square in the corner. The side of the triangle opposite the 90 ° 90 ° angle is called the hypotenuse and each of the other sides are called legs .

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, a 2 + b 2 = c 2 , a 2 + b 2 = c 2 , where a and b a and b are the lengths of the legs and c c is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.

The Pythagorean Theorem

In any right triangle, a 2 + b 2 = c 2 . a 2 + b 2 = c 2 .

where a and b are the lengths of the legs, c is the length of the hypotenuse.

To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation m m and the definition:

If m = n 2 , m = n 2 , then m = n , m = n , for n ≥ 0 . n ≥ 0 .

For example, we found that 25 25 is 5 because 25 = 5 2 . 25 = 5 2 .

Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

Example 3.39

Use the Pythagorean Theorem to find the length of the hypotenuse shown below.

Try It 3.77

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

Try It 3.78

Example 3.40.

Use the Pythagorean Theorem to find the length of the leg shown below.

Try It 3.79

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

Try It 3.80

Example 3.41.

Kelvin is building a gazebo and wants to brace each corner by placing a 10 ″ 10 ″ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

Try It 3.81

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

Try It 3.82

Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L , and its adjacent side as the width, W .

The distance around this rectangle is L + W + L + W , L + W + L + W , or 2 L + 2 W . 2 L + 2 W . This is the perimeter , P , of the rectangle.

What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

The area is the length times the width.

The formula for the area of a rectangle is A = L W . A = L W .

Properties of Rectangles

Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

The area of a rectangle is the product of the length and the width.

Example 3.42

The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

Try It 3.83

The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

Try It 3.84

The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

Example 3.43

The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?

Try It 3.85

The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

Try It 3.86

The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

Example 3.44

Find the length of a rectangle with perimeter 50 inches and width 10 inches.

Try It 3.87

Find the length of a rectangle with: perimeter 80 and width 25.

Try It 3.88

Find the length of a rectangle with: perimeter 30 and width 6.

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example 3.45

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

Try It 3.89

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

Try It 3.90

The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.

Example 3.46

The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Find the length and width.

Try It 3.91

The length of a rectangle is eight more than twice the width. The perimeter is 64. Find the length and width.

Try It 3.92

The width of a rectangle is six less than twice the length. The perimeter is 18. Find the length and width.

Example 3.47

The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

Try It 3.93

The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

Try It 3.94

The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

Section 3.4 Exercises

Practice makes perfect.

Solving Applications Using Triangle Properties

In the following exercises, solve using triangle properties.

The measures of two angles of a triangle are 26 and 98 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 105 and 31 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 47 and 72 degrees. Find the measure of the third angle.

The perimeter of a triangular pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?

A triangular courtyard has perimeter 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

If a triangle has sides 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?

If a triangle has sides 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, how long is the third side?

A triangular flag has base one foot and height 1.5 foot. What is its area?

A triangular window has base eight feet and height six feet. What is its area?

What is the base of a triangle with area 207 square inches and height 18 inches?

What is the height of a triangle with area 893 square inches and base 38 inches?

One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 51 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 22.5 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 36.5 degrees. What is the measure of the other small angle?

The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

One side of a triangle is twice the shortest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

One side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

The measure of the smallest angle of a right triangle is 20° less than the measure of the next larger angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is 20° more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.

In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.

A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display, as shown below. How far from the base of the pole should the end of the string of lights be anchored?

Pam wants to put a banner across her garage door, as shown below, to congratulate her son for his college graduation. The garage door is 12 feet high and 16 feet wide. How long should the banner be to fit the garage door?

Chi is planning to put a path of paving stones through her flower garden, as shown below. The flower garden is a square with side 10 feet. What will the length of the path be?

Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, how far up will the top of the ladder reach?

In the following exercises, solve using rectangle properties.

The length of a rectangle is 85 feet and the width is 45 feet. What is the perimeter?

The length of a rectangle is 26 inches and the width is 58 inches. What is the perimeter?

A rectangular room is 15 feet wide by 14 feet long. What is its perimeter?

A driveway is in the shape of a rectangle 20 feet wide by 35 feet long. What is its perimeter?

The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?

The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

Find the length of a rectangle with perimeter 124 and width 38.

Find the width of a rectangle with perimeter 92 and length 19.

Find the width of a rectangle with perimeter 16.2 and length 3.2.

Find the length of a rectangle with perimeter 20.2 and width 7.8.

The length of a rectangle is nine inches more than the width. The perimeter is 46 inches. Find the length and the width.

The width of a rectangle is eight inches more than the length. The perimeter is 52 inches. Find the length and the width.

The perimeter of a rectangle is 58 meters. The width of the rectangle is five meters less than the length. Find the length and the width of the rectangle.

The perimeter of a rectangle is 62 feet. The width is seven feet less than the length. Find the length and the width.

The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.

The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

The length of a rectangle is three times the width. The perimeter of the rectangle is 72 feet. Find the length and width of the rectangle.

The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

The length of a rectangle is five inches more than twice the width. The perimeter is 34 inches. Find the length and width.

The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Find the length and width of the rug.

Everyday Math

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other length?

Writing Exercises

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

Look at the two figures below.

• ⓐ Which figure looks like it has the larger area?
• ⓑ Which looks like it has the larger perimeter?
• ⓒ Now calculate the area and perimeter of each figure.
• ⓓ Which has the larger area?
• ⓔ Which has the larger perimeter?

Write a geometry word problem that relates to your life experience, then solve it and explain all your steps.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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• Published: 02 April 2024

A GAN-based genetic algorithm for solving the 3D bin packing problem

• Boliang Zhang 1 ,
• H. K. Kan 2 &
• Wuman Luo 1  

Scientific Reports volume  14 , Article number:  7775 ( 2024 ) Cite this article

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• Aerospace engineering
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The 3D bin packing problem is a challenging combinatorial optimization problem with numerous real-world applications. In this paper, we present a novel approach for solving this problem by integrating a generative adversarial network (GAN) with a genetic algorithm (GA). Our proposed GAN-based GA utilizes the GAN to generate high-quality solutions and improve the exploration and exploitation capabilities of the GA. We evaluate the performance of the proposed algorithm on a set of benchmark instances and compare it with two existing algorithms. The simulation studies demonstrate that our proposed algorithm outperforms both existing algorithms in terms of the number of used bins while achieving comparable computation times. Our proposed algorithm also performs well in terms of solution quality and runtime on instances of different sizes and shapes. We conduct sensitivity analysis and parameter tuning simulations to determine the optimal values for the key parameters of the proposed algorithm. Our results indicate that the proposed algorithm is robust and effective in solving the 3D bin packing problem. The proposed GAN-based GA algorithm and its modifications can be applied to other optimization problems. Our research contributes to the development of efficient and effective algorithms for solving complex optimization problems, particularly in the context of logistics and manufacturing. In summary, the proposed algorithm represents a promising solution to the challenging 3D bin packing problem and has the potential to advance the state-of-the-art in combinatorial optimization.

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Introduction

The 3D bin packing problem is a classical optimization problem in logistics and manufacturing, where a set of three-dimensional items needs to be packed into a minimum number of containers subject to certain constraints. The problem is known to be NP-hard, and finding an optimal solution is computationally challenging, especially for large problem instances 1 , 2 , 3 . Over the years, various optimization algorithms have been proposed to tackle this problem, including genetic algorithms (GA), simulated annealing, ant colony optimization, and particle swarm optimization. However, these algorithms suffer from limitations in terms of solution quality, convergence speed, or scalability 4 , 5 . Some advanced research works are shown in Table  1 and compared with our work.

The 3D bin packing problem is classified as NP-hard 12 , 13 , indicating the substantial computational complexity involved. This strong NP-hard nature of the problem implies that approximating the optimal solution within a constant factor is a significant challenge, unless the P = NP conjecture is disproven. Therefore, heuristic and metaheuristic algorithms are usually used to solve the problem in practice 14 . However, the existing algorithms for the 3D bin packing problem suffer from several challenges and limitations. First, some algorithms are not able to find optimal or near-optimal solutions for large problem instances due to their high computational complexity. Second, some algorithms are sensitive to the initial solution and are prone to get stuck in local optima. Third, some algorithms do not consider the complexity of the problem in terms of the number of items, bin dimensions, and volume limits, and may perform poorly on certain problem instances. Finally, some algorithms are not scalable and do not perform well on parallel computing architectures. These challenges and limitations motivate the need for new and improved optimization algorithms for the 3D bin packing problem 15 .

Recently, generative adversarial networks (GANs) have emerged as a powerful tool for generating high-quality synthetic data and solving optimization problems. GANs consist of two neural networks, a generator, and a discriminator, that are trained in an adversarial manner to generate realistic data samples 16 . Moreover, GANs have been applied to a variety of optimization problems, including image synthesis, text generation, and reinforcement learning. Specifically, Wu et al. 17 explored the potential of GANs in generating and reconstructing 3D objects, showcasing the versatility of GANs in handling three-dimensional data. On the other hand, GAs have been employed in various domains, from optimizing robot navigation in unstructured terrains 18 to enhancing space and time allocation in shipyard assembly halls 19 . Additionally, Tsai et al. 20 proposed a global optimization approach using GAs to solve three-dimensional open dimension rectangular packing problems, further emphasizing the adaptability and efficiency of GAs in solving spatial problems. GAs are optimization techniques inspired by natural evolution, widely applied in experimental designs as highlighted by the authors of 21 . On the other hand, quantum particle swarm optimization (QPSO), a variant of PSO with quantum mechanics principles, has been explored by the authors of 22 for its applications in designs with mixed factors and binary responses. Both methodologies offer innovative approaches to complex experimental design challenges.

The research objective of this paper is to propose a modified GA based on GANs to solve the 3D bin packing problem 23 , 24 , 25 . The proposed algorithm aims to combine the strengths of GA and GANs to offer a novel solution approach to the problem. In this paper, we present the mathematical formulation of the 3D bin packing problem and discuss the challenges and limitations of existing optimization algorithms. We then introduce our proposed algorithm, which includes a GAN-based modification, a new encoding scheme, a novel selection strategy, and a hybrid crossover and mutation operator. We also describe the design of the discriminator network, the training process, the fitness function, and the algorithm flowchart. Specifically, the proposed algorithm has the potential to address some of the key challenges faced by existing optimization algorithms. For instance, the GAN-based modification can help to generate high-quality candidate solutions, which can improve the overall solution quality. Additionally, the novel selection strategy and hybrid crossover and mutation operator can help to enhance the convergence speed of the algorithm, making it more efficient. The proposed algorithm can also be adapted to address other variants of the bin packing problem, such as the multiple knapsack problem and the strip packing problem. These problems have important applications in various industries, including transportation, logistics, and warehousing. Thus, the proposed algorithm has the potential to make a significant impact in these domains.

Overall, the proposed algorithm offers a promising approach for solving the 3D bin packing problem and has several potential applications in other combinatorial optimization problems. The successful application of GANs in optimization problems represents a significant step toward the development of more powerful and efficient optimization algorithms 26 . The proposed algorithm could open up new avenues for research in the fields of artificial intelligence and operations research, and pave the way for the development of more advanced optimization techniques. In addition, we conduct simulations on a set of benchmark instances and compare the performance of our proposed algorithm with existing state-of-the-art algorithms. The simulation studies demonstrate that our proposed algorithm outperforms the other methods in terms of solution quality and convergence speed.

The contributions of this paper include a novel approach to solving the 3D bin packing problem based on the combination of GA and GANs. The simulation validation of the effectiveness of our proposed algorithm. The implications of our proposed algorithm go beyond the 3D bin packing problem and could be a promising research area in the field of artificial intelligence and operations research.

The structure of this paper is as follows. “ Problem description ” describes the problem in this work. The proposed method is provided in “ Proposed method ”, followed by the simulation study in “ Simulation study ”. Discussion and analysis for the proposed algorithm are provided in “ Discussion ” and finally conclusions are drawn in “ Conclusion ”.

Problem description

The 3D bin packing problem can be formulated as follows 27 , 28 : given a set of n three-dimensional items, each with width \(w_i\) , height \(h_i\) , and depth \(d_i\) , and a set of identical three-dimensional bins, each with a fixed width W , height H , and depth D , the objective is to find a packing assignment that minimizes the number of bins used subject to the following constraints: (1) each item can only be packed once; (2) the total volume of the packed items in each bin cannot exceed the volume of the bin; (3) the orientation of each item is fixed, and it cannot be rotated or reflected. In other words, the problem can be formulated as an integer programming problem:

which subject to:

where m represents the total number of bins available for the bin-packing process. \(y_j\) is a binary decision variable. It equals 1 if bin j is used and 0 if it isn’t. \(x_{ij}\) is a binary variable that indicates whether item i is packed into bin j , and \(z_j\) is a binary variable that indicates whether bin j is used or not. The objective function minimizes the total number of used bins, and the constraints ensure that the items are packed into the bins without violating their dimensions or volume limits.

To better represent realistic scenarios, we minimize objective Z using the following constraints and variables:

where \(c_j\) denotes the capacity of bin j . \(p_j\) indicates the priority of bin j . Bins with higher priorities may be preferred based on the problem’s stipulations.

The objective function has been modified to include the minimization of a linear combination of the number of bins used \(\left( c_j z_j\right) \) and the processing time \(\left( p_j y_j\right) \) . The capacity constraints for weight, height, and depth are modified to include the binary variable \(z_j\) which indicates if \({\text {bin}} j\) is used. The variable \(y_j\) represents the processing time for bin j . Additional constraint \(\sum _{j=1}^m t_j y_j \le T\) ensures that the total processing time does not exceed a threshold T . The new constraint \(\sum _{i=1}^n r_i x_{i j} \le R z_j\) is added to account for an additional resource limitation represented by R . This revised formulation includes additional constraints that make the model more comprehensive and applicable to a broader set of real-world scenarios. The added variables and constraints take into account processing times and resource limitations, which are often critical factors in practical applications.

Proposed method

Theoretical basis.

Basic of GAN: it consist of two neural networks, namely the Generator and the Discriminator, that are trained simultaneously through adversarial processes. In the context of the 3D bin packing problem, the generator aims to produce realistic packing solutions while the Discriminator evaluates them. The Generator network, G , takes a random noise vector, z , as input and outputs a synthetic packing solution, G(z) . The architecture includes fully connected layers, and we use the ReLU activation function for the hidden layers. The output layer uses the sigmoid activation function to generate values between 0 and 1, which represent the assignment of items to bins. The Discriminator network, D , takes a packing solution (real or synthetic) as input and outputs a scalar representing the authenticity of the input. D(x) is close to 1 if x is a real packing solution and close to 0 if x is synthetic. The Discriminator is also a fully connected network, with the Leaky ReLU activation function in the hidden layers.

GAN-based modification: the cornerstone of our algorithm lies in the application of a generative adversarial network (GAN). In this network, two models are trained simultaneously: a generator model G , and a discriminator model D . The generator model G takes in random noise vector as input and generates synthetic packing assignments. The discriminator model D takes in both real and synthetic packing assignments and is tasked with distinguishing the former from the latter. As the training process progresses, the generator becomes increasingly proficient at producing packing assignments that closely resemble real ones, while the discriminator’s ability to distinguish real assignments from generated ones also improves. This symbiotic relationship facilitates the creation of realistic and diverse packing assignments.

Encoding scheme: our encoding scheme is pivotal for transforming complex packing assignments into manageable representations. We utilize a string of binary values, where each bit signifies whether an item is packed into a bin or not. The length of this string corresponds to the product of the total number of items and bins. This encoding scheme ensures the feasibility of the packing assignment while respecting the constraints of the problem.

Selection strategy: the selection strategy is a method for selecting individuals from the population for reproduction based on their fitness values. We use tournament selection, which selects a random subset of individuals from the population and chooses the fittest individual from the subset to be a parent.

Crossover and mutation operators: the crossover operator is a method for generating new individuals by combining the genetic material of two-parent individuals. We use a two-point crossover operator, which selects two random points in the encoding string and swaps the bits between the two points between the parents. The mutation operator is a method for introducing diversity into the population by randomly flipping bits in the encoding string.

Design details and training

Discriminator network design: the discriminator network is a neural network that distinguishes between real and synthetic packing assignments. The discriminator network consists of several layers of fully connected neurons, with the input layer taking as input the encoding string of the packing assignment and the output layer producing a single scalar value between 0 and 1, indicating the probability of the assignment being real.

The binary cross-entropy loss function used in the discriminator training phase is defined as follows:

where m is the batch size, \(x_i\) is the encoding string of the i -th packing assignment, \(y_i\) is the binary label indicating whether the assignment is real or synthetic, and \(D(x_i)\) is the output of the discriminator network for the i -th assignment.

Training process: training for our GAN-based algorithm occurs in two stages: generator training and discriminator training. In the generator training phase, we aim to improve the generator network’s ability to create feasible packing assignments by maximizing the feedback from the discriminator network. In the discriminator training phase, the goal is to better equip the discriminator network to differentiate between real and synthetic packing assignments. This is achieved by minimizing the binary cross-entropy loss function as follows:

where \(z_i\) is a random noise vector sampled from a normal distribution, and \(G(z_i)\) is the output of the generator network for the noise vector \(z_i\) . This loss function encourages the generator network to produce packing assignments that are similar to real assignments and that fool the discriminator network.

In the discriminator training phase, the discriminator network is trained to distinguish between real and synthetic packing assignments by minimizing the binary cross-entropy loss function defined in Eq. ( 1 ).

The training process for the GAN-based modification is shown in Fig.  1 . In each iteration, the generator network and the discriminator network are updated using a batch of real and synthetic data samples. The real data samples are randomly selected from the population of packing assignments, while the synthetic data samples are generated by the generator network using random noise vectors. The feedback from the discriminator network is used to update the generator network by backpropagating the gradients of the loss function for the generator parameters. The discriminator network is updated by backpropagating the gradients of the loss function for the discriminator parameters.

Training process for the GAN-based modification.

Fitness function: the fitness function is a measure of the quality of a packing assignment. We use the number of used bins as the fitness value, which is the objective of the 3D bin packing problem. The fitness function ensures that the fittest individuals are selected for reproduction and that the algorithm converges to a near-optimal solution.

Algorithm flowchart: the algorithm flowchart is shown in Fig.  2 . The algorithm starts by initializing the population using the encoding scheme. The algorithm then evaluates the fitness of each individual using the fitness function. The algorithm then enters the main loop, which consists of the following steps: selection, crossover, mutation, evaluation, and GAN-based modification. The algorithm stops when a stopping criterion is met, such as a maximum number of generations or a satisfactory fitness value.

Integration GANs with the GA: the integration of GANs within the GA is carried out in two primary ways as follows.

Diversity injection: after every \(k\) generations of the GA, we use \(G\) to generate synthetic solutions, which are introduced into the population.

Fitness augmentation: the discriminator \(D\) is used to calculate an auxiliary fitness component for the individuals, measuring the solution’s authenticity.

This leads to the augmented fitness function:

where \(F\) is the original fitness, \(F'\) is the augmented fitness, \(D(x)\) is the discriminator’s output, and \(\alpha \) is a weighting factor.

Algorithm flowchart.

The interplay between GAN and GA.

Algorithm design

Our proposed algorithm, as depicted in Algorithm 1, is a modified GA incorporating GANs to address the challenges of the 3D bin packing problem. In this algorithm, we utilize various components and strategies to optimize the population over generations. At its core, P ( t ) represents the population at generation t , with G denoting the GAN generator and D the GAN discriminator. The synthetic population, denoted as \(P_s(t)\) , is generated by the GAN. The combined population of real and synthetic individuals at generation t is represented by \(P^{\prime }(t)\) . Subsequently, we perform crossover operations, resulting in population \(P^{\prime \prime }(t)\) , followed by mutation operations to obtain population \(P^{\prime \prime \prime }(t)\) . The fitness function, denoted as f ( x ), is evaluated for each individual x in the population, guiding the selection process. Our algorithm integrates the strengths of both GANs and GAs to enhance the optimization process. The GAN primarily functions to generate high-quality solutions that emulate optimal or near-optimal solutions, contributing to diversifying the population and introducing potentially superior individuals. Additionally, the GAN aids in increasing population diversity. In contrast, the GA serves as an exploitation mechanism, evolving the population by selecting the best individuals based on their fitness and employing crossover and mutation operators. The interplay between the GAN and GA components is illustrated in Fig.  3 . This combination allows our algorithm to effectively address the limitations of existing optimization algorithms for the 3D bin packing problem, ultimately leading to improved results. Specifically, the proposed algorithm can be divided into following steps:

The first step in our GAN-based GA algorithm involves initializing a population of individuals with randomly generated packing assignments. We chose a binary representation for the individuals, where each bit represents whether an item is packed into a bin or not. This simple representation was chosen for its ease of implementation and interpretability.

GAN-based GA for 3D bin packing.

The GAN component of our algorithm consists of a generator and a discriminator network. The generator is tasked with generating feasible packing assignments from random noise vectors, while the discriminator’s role is to distinguish between real and synthetic packing assignments. The training process involves updating the generator network using the feedback from the discriminator network, while the discriminator network is updated based on its ability to distinguish between real and synthetic packing assignments. We used the binary cross-entropy loss function for both networks, as it is commonly used in GANs for its robustness and ease of optimization.

In the genetic operations stage, we used tournament selection due to its simplicity and effectiveness. For crossover, we employed a two-point crossover operator, which is a common choice in GA literature. The mutation was applied by randomly flipping bits in the binary representation of the individuals, ensuring diversity in our population.

The discriminator network design followed a simple architecture of fully connected layers with binary output, indicating the probability of a packing assignment being real. The choice of a simple architecture was made to reduce the complexity of the model, improve training speed, and avoid overfitting.

The feedback loop between the discriminator and generator networks was implemented using backpropagation, a standard method in neural network training. Gradients were computed based on the loss function and used to update the parameters of the two networks.

In the context of aviation, the proposed algorithm can have significant implications for optimizing airport logistics. With the rise of smart airports and the Internet of Things (IoT), there is a growing need for efficient and effective optimization techniques to handle the complex logistics involved in airport operations. The proposed algorithm can be a valuable tool in this context by providing an automated and intelligent approach to packing cargo and luggage, reducing handling time and increasing efficiency.

Time complexity analysis

The time complexity of our proposed algorithm is composed of the complexities of several key steps: the generator and discriminator network training in the GAN component, the genetic operation steps (selection, crossover, mutation), and the evaluation of the fitness function. In the worst-case scenario, the time complexity of our algorithm can be represented as O( N ), where N denotes the number of generations times the population size times the complexity of the discriminator and generator training. However, it is important to note that our proposed algorithm, while may be higher in time complexity compared to traditional GAs and PSOs due to the addition of the GAN component, often converges to a solution faster due to more efficient exploration and exploitation of the search space.

Simulation study

To evaluate the performance of our proposed algorithm, we used benchmark instances from the literature on 3D bin packing. The instances were generated using a variety of problem sizes, ranging from small instances with 10 items and 3 bins to large instances with 100 items and 20 bins. Table  2 provides specific information for each instance, such as the number of items, their respective dimensions, the number of bins and their capacity. We randomly generated the dimensions and weights of the items and the dimensions of the bins, ensuring that the instances were feasible and realistic. Table  3 provides a concise overview of three distinct optimization algorithms employed for addressing the 3D bin packing problem. The algorithms include the GAN-based GA, Conventional GA, and PSO. The Table  3 lists key parameters for each algorithm, encompassing the number of evaluations, average time per evaluation, total computation time, population or swarm size, crossover rate, mutation rate, GAN training interval, GAN epochs per interval, as well as the cognitive and social coefficients in the PSO algorithm.

We compared the performance of our proposed algorithm with two existing algorithms for the 3D bin packing problem: a traditional GA and a state-of-the-art algorithm based on PSO. We used the same instances and simulation setup for all three algorithms to ensure a fair comparison.

In the process of the simulations, we ran each algorithm for 50 generations with a population size of 100 individuals. We repeated each simulation 10 times and reported the average results. We also used a statistical test, the Wilcoxon signed-rank test, to determine whether the differences in the results were statistically significant.

Comparison and analysis

In this paper, two evaluation metrics are used to compare the performance of the algorithms: the number of used bins and the computation time. The number of used bins is the objective of the 3D bin packing problem and measures the efficiency of the packing assignment. The computation time measures the speed and efficiency of the algorithm. Moreover, the sensitivity analysis and parameter tuning simulations are conducted to investigate the effect of the key parameters on the performance of our proposed algorithm. We varied the population size, the number of generations, the mutation rate, and the GAN training parameters to determine the optimal values for each parameter.

As shown in Table  4 , the performance of our proposed algorithm is compared with the traditional GA and the PSO algorithm on the benchmark instances. It is clear that our proposed algorithm outperformed both GA and PSO in all instances in terms of the number of used bins. The average improvement in the number of used bins over GA was \(9.1\%\) , while the average improvement over PSO was \(5.8\%\) . The differences in the results were statistically significant according to the Wilcoxon signed-rank test. In addtion, the computation time of our proposed algorithm was comparable to that of GA and PSO. The average computation time for the proposed algorithm was 10.2 s , while the average computation times for GA and PSO were 9.7 s and 11.5 s, respectively.

To evaluate the performance and stability of the GA, PSO, and GAN-based GHA algorithms, we ran multiple iterations of each algorithm and recorded the results. Table  5 presents the average performance metrics for each algorithm across different datasets. Additionally, the table includes the standard deviations, which provides insights into the variability of each algorithm’s performance. In addition to the previously mentioned algorithms, we have now compared our method with the following state-of-the-art algorithms:

Deep reinforcement learning for 3D-BPP (DRL-3D-BPP): this approach uses deep Q-learning to find efficient packing solutions.

Hybrid simulated annealing and Tabu search (SA-TS) for 3D-BPP: combines the global search of simulated annealing with the local search capabilities of tabu search.

As shown in Table  6 , our GAN-based GA approach consistently outperformed the DRL-3D-BPP in terms of solution quality, obtaining on average a 7% reduction in the number of bins required. When compared to the SA-TS hybrid method, our algorithm demonstrated a 5% improvement. Moreover, the computation time was also competitive. While the DRL-3D-BPP was faster by a margin of 10%, our method was more efficient than the SA-TS by approximately 15%. It’s worth noting that while our method’s absolute speed might not always surpass every latest algorithm, the quality of solutions and consistency it delivers makes it a formidable approach for the 3D-BPP. The inclusion of the GAN mechanism enables our GA to maintain diversity and avoid premature convergence. This factor plays a significant role in the algorithm’s ability to consistently find near-optimal solutions, giving it an edge over certain latest methodologies. Furthermore, our analysis suggests that the collaboration between GAN and GA in our approach offers a robust balance between exploration and exploitation. This balance, in turn, provides a compelling argument for its suitability in addressing challenging combinatorial optimization problems like 3D-BPP.

Extended simulation

To better verify the performance of the proposed method, we have broadened the datasets used in the simulations. The proposed algorithm has been tested on a total of seven datasets, namely Pack1 to Pack7. These datasets are subsets of the OR-Library, which contains various benchmark instances for the 3D bin packing problem and is widely used in the research community. Employing this dataset allows for a more robust and comparative analysis of our method’s performance.

We have extended the comparison of our proposed algorithm with more recent methods found in the literature, as shown in Table  7 . Specifically, we added three more comparison methods: BB-BC (branched and bounded-best fit decreasing), FFD (first-fit decreasing), and VPS (variable partition search). These methods are acknowledged as state-of-the-art and have been extensively applied in the bin-packing problem, making them appropriate choices for performance comparison.

Regarding the assumptions in our simulations, all the tests were conducted under the same conditions for all algorithms for fairness. All parameters for each algorithm were fine-tuned for best performance on the given datasets. The variation of input parameters for our proposed algorithm has been thoroughly investigated in the sensitivity analysis section of the paper.

Multi-factors sensitive analysis

Table  8 presents the detailed results of the multi-factor sensitivity analysis. For each combination of parameters, we recorded the average number of used bins and the average computation time over 30 runs. The standard deviation measures the variability of the results. As observed from Table  8 , the algorithm’s performance is significantly influenced by the interaction between the population size, the number of generations, and the mutation rate. A detailed analysis and discussion on the specific effects of these interactions will be provided in the next section.

Figure  4 demonstrate that the variability of our proposed algorithm in performance across different parameter settings, the GAN-based modification tends to buffer some of the extreme sensitivities observed in traditional GAs. Compared to PSO’s robustness over parameter changes, our proposed algorithm’s sensitivity lies in between traditional GA and PSO, which is attributed to the GAN component. The red bars represent the performance measure (fitness value) for each parameter setting. The blue line plot indicates the convergence time for each setting.

From the above results, it’s evident that the performance of our algorithm varies with different parameter settings, particularly with changes in crossover rate and mutation rate. However, these fluctuations in performance are not as pronounced as one might observe in a purely GA-based approach. For instance, while the fitness value peaks at a crossover rate of 0.75, it does not plummet drastically when the rate is altered slightly, demonstrating the resilience added by our GAN component. Similarly, changes in the mutation rate affect performance, but the presence of the GAN-based modification ensures that the sensitivity is within a manageable range. When we compare this behavior with the robustness of PSO over parameter changes, our proposed algorithm, equipped with the GAN component, seems to strike a balance between the high sensitivity observed in traditional GA and the robustness of PSO.

Impact of parameters on performance measure and convergence time.

Figures  5 and   6 further analyzed the performance of the algorithms using the two performance evaluation metrics, which show the comparisons of the number of used bins and the computation time for the three algorithms. As shown in Fig.  5 , the proposed algorithm achieved the best results in terms of the number of used bins in all instances. The median and interquartile range (IQR) of the number of used bins for the proposed algorithm were consistently lower than those of GA and PSO. Figure  6 shows that the computation time of our proposed algorithm was comparable to that of GA and PSO. The median and IQR of the computation time for the proposed algorithm were slightly higher than those of GA but lower than those of PSO.

Comparison of the number of used bins for the proposed algorithm, GA, and PSO.

Comparison of the number of the computation time for the proposed algorithm, GA, and PSO.

Figures  7 and   8 show the results of the sensitivity analysis for the number of used bins and the computation time. As shown in Fig.  7 , the number of used bins decreased as the population size increased, up to a certain point where the improvement plateaued. The optimal population size was found to be 200, which resulted in an average improvement of 12.5% over the baseline. Figure  8 shows that the computation time increased as the population size increased, but the increase was relatively small. The optimal population size was found to be 200, which resulted in an average computation time of 11.3 s.

Results of the sensitivity analysis for the number of used bins with various population size.

Results of the sensitivity analysis for the computation time with various population size.

Figures  9 and   10 show the results of the sensitivity analysis for the number of used bins and the computation time for the number of generations. As shown in Fig.  9 , the number of used bins improved as the number of generations increased, up to a certain point where the improvement plateaued. The optimal number of generations was found to be 100, which resulted in an average improvement of 11.8% over the baseline. Figure  10 shows that the computation time increased as the number of generations increased, but the increase was relatively small. The optimal number of generations was found to be 100, which resulted in an average computation time of 11.0 s.

Results of the sensitivity analysis for the number of used bins with various the number of generations.

Results of the sensitivity analysis for the computation time with various the number of generations.

Figures  11 and   12 display the results of the sensitivity analysis for the number of used bins and the computation time for the mutation rate. As shown in Fig.  11 , the number of used bins improved as the mutation rate increased, up to a certain point where the improvement plateaued. The optimal mutation rate was found to be 0.05, which resulted in an average improvement of 11.1% over the baseline.

Figure  12 shows that the computation time increased as the mutation rate increased, but the increase was relatively small. The optimal mutation rate was found to be 0.05, which resulted in an average computation time of 11.1 s.

Results of the sensitivity analysis for the number of used bins with various mutation rate.

Results of the sensitivity analysis for the computation time with various mutation rate.

Finally, we conducted a parameter tuning simulation to determine the optimal GAN training parameters, including the learning rate, the batch size, and the number of training iterations. The optimal values for these parameters were found to be 0.0002, 32, and 5000, respectively. These parameters resulted in an average improvement of 11.3% over the baseline and an average computation time of 11.2 s.

In summary, our simulation results show that the proposed algorithm outperformed both GA and PSO on the benchmark instances in terms of the number of used bins. The proposed algorithm also achieved comparable computation times to GA and PSO. The sensitivity analysis and parameter tuningsimulations revealed the optimal values for the key parameters of the proposed algorithm, including the population size, the number of generations, the mutation rate, and the GAN training parameters.

Interpretation of the results

Our simulation results showed that the proposed algorithm, which is a modified genetic algorithm based on GANs, outperformed both the traditional GA and the PSO algorithm on the benchmark instances for the 3D bin packing problem. The proposed algorithm achieved an average improvement of 9.1% and 5.8% over GA and PSO, respectively, in terms of the number of used bins. The computation time of the proposed algorithm was also comparable to that of GA and PSO.

Furthermore, the sensitivity analysis and parameter tuning simulations revealed the optimal values for the key parameters of the proposed algorithm, including the population size, the number of generations, the mutation rate, and the GAN training parameters. The optimal values resulted in an average improvement of 11.3% over the baseline and an average computation time of 11.2 s.

Analysis of the effectiveness of the GAN-based modification

The GAN-based modification was designed to improve the diversity and quality of the population by generating realistic and diverse packing assignments. Our simulation results showed that the GAN-based modification was effective in improving the performance of the proposed algorithm, especially in terms of the number of used bins. The GAN-based modification allowed the algorithm to generate more diverse packing assignments that were closer to the optimal solution. The discriminator network was effective in distinguishing between real and synthetic packing assignments and providing feedback to the generator network to improve its performance.

Lmitations and potential improvements

Although the proposed algorithm avoids the problem that existing algorithms easily fall into local optimal solutions, it still relies on the quality of the initial solution. The algorithm may perform poorly if the initial population is not diverse or does not contain high-quality individuals. One potential improvement is to use a hybrid initialization method that combines different encoding schemes or other optimization techniques to generate a diverse and high-quality initial population.

Another limitation is the sensitivity of the algorithm to the choice of parameters. As shown in Fig.  4 , the performance of the algorithm may vary significantly with different parameter settings, and finding the optimal values may require a significant amount of time and computational resources. One potential improvement is to use a more efficient and automated parameter tuning method, such as Bayesian optimization or reinforcement learning.

Implications for future research

Our proposed algorithm demonstrated the effectiveness of using GANs to improve the performance of genetic algorithms for the 3D bin packing problem. This opens up a new direction for future research on using GANs in other optimization problems or combining GANs with other metaheuristic algorithms. The sensitivity analysis and parameter tuning simulations also highlight the importance of parameter tuning and optimization in designing effective algorithms for optimization problems. Future research could investigate more efficient and automated parameter tuning methods or develop new optimization techniques that are less sensitive to parameter choices.

In this paper, we presented a modified GA based on GANs for the 3D bin packing problem. Our proposed algorithm utilized the GAN-based modification to improve the diversity and quality of the population and outperformed traditional GA and PSO algorithms on benchmark instances. The simulation studies demonstrated that the proposed algorithm achieved better performance than the baseline algorithms in terms of the number of used bins while maintaining comparable computation times. The sensitivity analysis and parameter tuning simulations revealed the optimal values for the key parameters of the proposed algorithm, including the population size, the number of generations, the mutation rate, and the GAN training parameters. The GAN-based modification was effective in improving the diversity and quality of the population and generating realistic and diverse packing assignments. In summary, our proposed algorithm demonstrates the effectiveness of using GANs to improve the performance of genetic algorithms for the 3D bin packing problem and opens up new avenues for future research in the field of optimization. The proposed algorithm and its modifications can be applied to other optimization problems, and the GAN-based approach can be used to generate synthetic data in various domains, including the aviation industry, where data acquisition can be challenging and expensive.

Data availability

The datasets generated and/or analysed during the current study are available in the GitHub repository, https://github.com/wjszbl/3DGAPA .

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Acknowledgements

This work is supported in part by the research Grant (No.: RP/ESCA-07/2021) offered by Macao Polytechnic University.

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B.Z.; methodology, B.Z.; software, B.Z.; validation, B.Z.; formal analysis, B.Z. and Y.Y.; investigation, B.Z., Y.Y., and W.L.; resources, Y.Y., H.K, Kan, and W.L.; data curation,Y.Y.; writing-original draft preparation, Y.Y.; writing-review and editing, Y.Y., H.K, Kan, and W.L.; visualization, B.Z. and Y.Y.; supervision, H.K, Kan.; project administration, H.K, Kan and W.L.; funding acquisition, H.K, Kan. All authors have read and agreed to the published version of the manuscript.

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Zhang, B., Yao, Y., Kan, H.K. et al. A GAN-based genetic algorithm for solving the 3D bin packing problem. Sci Rep 14 , 7775 (2024). https://doi.org/10.1038/s41598-024-56699-7

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1.5: Applications of Linear Systems

• Last updated
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• Page ID 63381

• Gregory Hartman et al.
• Virginia Military Institute

Learning Objectives

• How do most problems appear “in the real world?”
• The unknowns in a problem are also called what?
• How many points are needed to determine the coefficients of a 5th degree polynomial?

We’ve started this chapter by addressing the issue of finding the solution to a system of linear equations. In subsequent sections, we defined matrices to store linear equation information; we described how we can manipulate matrices without changing the solutions; we described how to efficiently manipulate matrices so that a working solution can be easily found.

We shouldn’t lose sight of the fact that our work in the previous sections was aimed at finding solutions to systems of linear equations. In this section, we’ll learn how to apply what we’ve learned to actually solve some problems.

Many, many, many problems that are addressed by engineers, businesspeople, scientists and mathematicians can be solved by properly setting up systems of linear equations. In this section we highlight only a few of the wide variety of problems that matrix algebra can help us solve.

We start with a simple example.

Example \(\PageIndex{1}\)

A jar contains \(100\) blue, green, red and yellow marbles. There are twice as many yellow marbles as blue; there are \(10\) more blue marbles than red; the sum of the red and yellow marbles is the same as the sum of the blue and green. How many marbles of each color are there?

Let’s call the number of blue balls \(b\) , and the number of the other balls \(g\) , \(r\) and \(y\) , each representing the obvious. Since we know that we have 100 marbles, we have the equation \[b+g+r+y=100. \nonumber \] The next sentence in our problem statement allows us to create three more equations.

We are told that there are twice as many yellow marbles as blue. One of the following two equations is correct, based on this statement; which one is it? \[2y=b \quad\quad \text{or} \quad\quad2b=y \nonumber \]

The first equation says that if we take the number of yellow marbles, then double it, we’ll have the number of blue marbles. That is not what we were told. The second equation states that if we take the number of blue marbles, then double it, we’ll have the number of yellow marbles. This is what we were told.

The next statement of “there are 10 more blue marbles as red” can be written as either \[b=r+10 \quad\quad \text{or} \quad\quad r=b+10. \nonumber \] Which is it?

The first equation says that if we take the number of red marbles, then add 10, we’ll have the number of blue marbles. This is what we were told. The next equation is wrong; it implies there are more red marbles than blue.

The final statement tells us that the sum of the red and yellow marbles is the same as the sum of the blue and green marbles, giving us the equation \[r+y=b+g. \nonumber \]

We have four equations; altogether, they are \[\begin{align}\begin{aligned} b+g+r+y&= 100\\ 2b&=y\\ b&=r+10\\ r+y&=b+g.\\ \end{aligned}\end{align} \nonumber \]

We want to write these equations in a standard way, with all the unknowns on the left and the constants on the right. Let us also write them so that the variables appear in the same order in each equation (we’ll use alphabetical order to make it simple). We now have \[\begin{align}\begin{aligned} b+g+r+y&=100\\2b-y&=0\\b-r&=10\\-b-g+r+y&=0\\ \end{aligned}\end{align} \nonumber \]

To find the solution, let’s form the appropriate augmented matrix and put it into reduced row echelon form. We do so here, without showing the steps.

\[\left[\begin{array}{ccccc}{1}&{1}&{1}&{1}&{100}\\{2}&{0}&{0}&{-1}&{0}\\{1}&{0}&{-1}&{0}&{10}\\{-1}&{-1}&{1}&{1}&{0}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{ccccc}{1}&{0}&{0}&{0}&{20}\\{0}&{1}&{0}&{0}&{30}\\{0}&{0}&{1}&{0}&{10}\\{0}&{0}&{0}&{1}&{40}\end{array}\right] \nonumber \]

We interpret from the reduced row echelon form of the matrix that we have 20 blue, 30 green, 10 red and 40 yellow marbles.

Even if you had a bit of difficulty with the previous example, in reality, this type of problem is pretty simple. The unknowns were easy to identify, the equations were pretty straightforward to write (maybe a bit tricky for some), and only the necessary information was given.

Most problems that we face in the world do not approach us in this way; most problems do not approach us in the form of “Here is an equation. Solve it.” Rather, most problems come in the form of:

Here is a problem. I want the solution. To help, here is lots of information. It may be just enough; it may be too much; it may not be enough. You figure out what you need; just give me the solution.

Faced with this type of problem, how do we proceed? Like much of what we’ve done in the past, there isn’t just one “right” way. However, there are a few steps that can guide us. You don’t have to follow these steps, “step by step,” but if you find that you are having difficulty solving a problem, working through these steps may help.

While the principles outlined here will help one solve any type of problem, these steps are written specifically for solving problems that involve only linear equations.

Key Idea \(\PageIndex{1}\): Mathematical Problem Solving

• Understand the problem. What exactly is being asked?
• Identify the unknowns. What are you trying to find? What units are involved?
• Give names to your unknowns (these are your variables).
• Use the information given to write as many equations as you can that involve these variables.
• Use the equations to form an augmented matrix; use Gaussian elimination to put the matrix into reduced row echelon form.
• Interpret the reduced row echelon form of the matrix to identify the soluƟon.
• Ensure the solution makes sense in the context of the problem.

Having identified some steps, let us put them into practice with some examples.

Example \(\PageIndex{2}\)

A concert hall has seating arranged in three sections. As part of a special promotion, guests will recieve two of three prizes. Guests seated in the first and second sections will receive Prize A, guests seated in the second and third sections will receive Prize B, and guests seated in the first and third sections will receive Prize C. Concert promoters told the concert hall managers of their plans, and asked how many seats were in each section. (The promoters want to store prizes for each section separately for easier distribution.) The managers, thinking they were being helpful, told the promoters they would need \(105\) A prizes, \(103\) B prizes, and \(88\) C prizes, and have since been unavailable for further help. How many seats are in each section?

Before we rush in and start making equations, we should be clear about what is being asked. The final sentence asks: “How many seats are in each section?” This tells us what our unknowns should be: we should name our unknowns for the number of seats in each section. Let \(x_1\) , \(x_2\) and \(x_3\) denote the number of seats in the first, second and third sections, respectively. This covers the first two steps of our general problem solving technique.

(It is tempting, perhaps, to name our variables for the number of prizes given away. However, when we think more about this, we realize that we already know this – that information is given to us. Rather, we should name our variables for the things we don’t know.)

Having our unknowns identified and variables named, we now proceed to forming equations from the information given. Knowing that Prize A goes to guests in the first and second sections and that we’ll need 105 of these prizes tells us \[x_1+x_2 = 105. \nonumber \] Proceeding in a similar fashion, we get two more equations, \[x_2+x_3 = 103\quad\text{ and }\quad x_1+x_3 = 88. \nonumber \] Thus our linear system is \[\begin{array}{rcl}x_1+x_2&=&105\\x_2+x_3&=&103\\x_1+x_3&=&88\\ \end{array} \nonumber \] and the corresponding augmented matrix is

\[\left[\begin{array}{cccc}{1}&{1}&{0}&{105}\\{0}&{1}&{1}&{103}\\{1}&{0}&{1}&{88}\end{array}\right] \nonumber \]

To solve our system, let’s put this matrix into reduced row echelon form.

\[\left[\begin{array}{cccc}{1}&{1}&{0}&{105}\\{0}&{1}&{1}&{103}\\{1}&{0}&{1}&{88}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{cccc}{1}&{0}&{0}&{45}\\{0}&{1}&{0}&{60}\\{0}&{0}&{1}&{43}\end{array}\right] \nonumber \]

We can now read off our solution. The first section has 45 seats, the second has 60 seats, and the third has 43 seats.

Example \(\PageIndex{3}\)

A lady takes a 2-mile motorized boat trip down the Highwater River, knowing the trip will take 30 minutes. She asks the boat pilot “How fast does this river flow?” He replies “I have no idea, lady. I just drive the boat.”

She thinks for a moment, then asks “How long does the return trip take?” He replies “The same; half an hour.” She follows up with the statement, “Since both legs take the same Ɵme, you must not drive the boat at the same speed.”

“Naw,” the pilot said. “While I really don’t know exactly how fast I go, I do know that since we don’t carry any tourists, I drive the boat twice as fast.”

The lady walks away satisfied; she knows how fast the river flows.

(How fast does it flow?)

This problem forces us to think about what information is given and how to use it to find what we want to know. In fact, to find the solution, we’ll find out extra information that we weren’t asked for!

We are asked to find how fast the river is moving (step 1). To find this, we should recognize that, in some sense, there are three speeds at work in the boat trips: the speed of the river (which we want to find), the speed of the boat, and the speed that they actually travel at.

We know that each leg of the trip takes half an hour; if it takes half an hour to cover 2 miles, then they must be traveling at 4 mph, each way.

The other two speeds are unknowns, but they are related to the overall speeds. Let’s call the speed of the river \(r\) and the speed of the boat \(b\) . (And we should be careful. From the conversation, we know that the boat travels at two different speeds. So we’ll say that \(b\) represents the speed of the boat when it travels downstream, so \(2b\) represents the speed of the boat when it travels upstream.) Let’s let our speed be measured in the units of miles/hour (mph) as we used above (steps 2 and 3).

What is the rate of the people on the boat? When they are travelling downstream, their rate is the sum of the water speed and the boat speed. Since their overall speed is 4 mph, we have the equation \(r+b=4\) .

When the boat returns going against the current, its overall speed is the rate of the boat minus the rate of the river (since the river is working against the boat). The overall trip is still taken at 4 mph, so we have the equation \(2b-r=4\) . (Recall: the boat is traveling twice as fast as before.)

The corresponding augmented matrix is

\[\left[\begin{array}{ccc}{1}&{1}&{4}\\{2}&{-1}&{4}\end{array}\right] \nonumber \]

Note that we decided to let the first column hold the coefficients of \(b\) .

Putting this matrix in reduced row echelon form gives us:

\[\left[\begin{array}{ccc}{1}&{1}&{4}\\{2}&{-1}&{4}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{ccc}{1}&{0}&{8/3}\\{0}&{1}&{4/3}\end{array}\right] \nonumber \]

We finish by interpreting this solution: the speed of the boat (going downstream) is 8/3 mph, or \(2.\overline{6}\) mph, and the speed of the river is 4/3 mph, or \(1.\overline{3}\) mph. All we really wanted to know was the speed of the river, at about 1.3 mph.

Example \(\PageIndex{4}\)

Find the equation of the quadratic function that goes through the points \((−1,\: 6)\), \((1,\: 2)\) and \((2,\: 3)\).

This may not seem like a “linear” problem since we are talking about a quadratic function, but closer examination will show that it really is.

We normally write quadratic functions as \(y=ax^2+bx+c\) where \(a\) , \(b\) and \(c\) are the coefficients; in this case, they are our unknowns. We have three points; consider the point \((-1,6)\) . This tells us directly that if \(x=-1\) , then \(y=6\) . Therefore we know that \(6=a(-1)^2+b(-1)+c\) . Writing this in a more standard form, we have the linear equation \[a - b+c=6. \nonumber \]

The second point tells us that \(a(1)^2+b(1)+c = 2\) , which we can simplify as \(a+b+c=2\) , and the last point tells us \(a(2)^2+b(2)+c = 3\) , or \(4a+2b+c=3\) . Thus our linear system is \[\begin{array}{rcl} a-b+c&=&6\\ a+b+c&=&2\\ 4a+2b+c&=&3.\\ \end{array} \nonumber \]

Again, to solve our system, we find the reduced row echelon form of the corresponding augmented matrix. We don’t show the steps here, just the final result.

\[\left[\begin{array}{cccc}{1}&{-1}&{1}&{6}\\{1}&{1}&{1}&{2}\\{4}&{2}&{1}&{3}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{cccc}{1}&{0}&{0}&{1}\\{0}&{1}&{0}&{-2}\\{0}&{0}&{1}&{3}\end{array}\right] \nonumber \]

This tells us that \(a=1\) , \(b=-2\) and \(c=3\) , giving us the quadratic function \(y=x^2-2x+3\) .

One thing interesting about the previous example is that it confirms for us something that we may have known for a while (but didn’t know why it was true). Why do we need two points to find the equation of the line? Because in the equation of the a line, we have two unknowns, and hence we’ll need two equations to find values for these unknowns.

A quadratic has three unknowns (the coefficients of the \(x^2\) term and the \(x\) term, and the constant). Therefore we’ll need three equations, and therefore we’ll need three points.

What happens if we try to find the quadratic function that goes through 3 points that are all on the same line? The fast answer is that you’ll get the equation of a line; there isn’t a quadratic function that goes through 3 colinear points. Try it and see! (Pick easy points, like \((0,0)\) , \((1,1)\) and \((2,2)\) . You’ll find that the coefficient of the \(x^2\) term is 0.)

Of course, we can do the same type of thing to find polynomials that go through 4, 5, etc., points. In general, if you are given \(n+1\) points, a polynomial that goes through all \(n+1\) points will have degree at most \(n\) .

Example \(\PageIndex{5}\)

A woman has 32 $1, $5 and $10 bills in her purse, giving her a total of $100. How many bills of each denomination does she have?

Let’s name our unknowns \(x\) , \(y\) and \(z\) for our ones, fives and tens, respectively (it is tempting to call them \(o\) , \(f\) and \(t\) , but \(o\) looks too much like 0). We know that there are a total of 32 bills, so we have the equation \[x+y+z = 32. \nonumber \] We also know that we have $100, so we have the equation \[x+5y+10z = 100. \nonumber \] We have three unknowns but only two equations, so we know that we cannot expect a unique solution. Let’s try to solve this system anyway and see what we get.

Putting the system into a matrix and then finding the reduced row echelon form, we have

\[\left[\begin{array}{cccc}{1}&{1}&{1}&{32}\\{1}&{5}&{10}&{100}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{cccc}{1}&{0}&{-\frac{5}{4}}&{15}\\{0}&{1}&{\frac{9}{4}}&{17}\end{array}\right] \nonumber \]

Reading from our reduced matrix, we have the infinite solution set \[\begin{align}\begin{aligned} x &=15+\frac54z\\ y&=17 - \frac94z\\ z & \text{ is free.}\end{aligned}\end{align} \nonumber \]

While we do have infinite solutions, most of these solutions really don’t make sense in the context of this problem. (Setting \(z = \frac12\) doesn’t make sense, for having half a ten dollar bill doesn’t give us $5. Likewise, having \(z = 8\) doesn’t make sense, for then we’d have “ \(-1\) ” $5 bills.) So we must make sure that our choice of \(z\) doesn’t give us fractions of bills or negative amounts of bills.

To avoid fractions, \(z\) must be a multiple of 4 ( \(-4, 0, 4, 8, \ldots\) ). Of course, \(z\geq 0\) for a negative number wouldn’t make sense. If \(z = 0\) , then we have 15 one dollar bills and 17 five dollar bills, giving us $100. If \(z = 4\) , then we have \(x = 20\) and \(y = 8\) . We already mentioned that \(z=8\) doesn’t make sense, nor does any value of \(z\) where \(z\geq 8\) .

So it seems that we have two answers; one with \(z=0\) and one with \(z=4\) . Of course, by the statement of the problem, we are led to believe that the lady has at least one $10 bill, so probably the “best” answer is that we have 20 $1 bills, 8 $5 bills and 4 $10 bills. The real point of this example, though, is to address how infinite solutions may appear in a real world situation, and how suprising things may result.

Example \(\PageIndex{6}\)

In a football game, teams can score points through touchdowns worth 6 points, extra points (that follow touchdowns) worth 1 point, two point conversions (that also follow touchdowns) worth 2 points and field goals, worth 3 points. You are told that in a football game, the two competing teams scored on 7 occasions, giving a total score of 24 points. Each touchdown was followed by either a successful extra point or two point conversion. In what ways were these points scored?

The question asks how the points were scored; we can interpret this as asking how many touchdowns, extra points, two point conversions and field goals were scored. We’ll need to assign variable names to our unknowns; let \(t\) represent the number of t ouchdowns scored; let \(x\) represent the number of e x tra points scored, let \(w\) represent the number of t w o point conversions, and let \(f\) represent the number of f ield goals scored.

Now we address the issue of writing equations with these variables using the given information. Since we have a total of 7 scoring occasions, we know that \[t+x+w+f=7. \nonumber \] The total points scored is 24; considering the value of each type of scoring opportunity, we can write the equation \[6t+x+2w+3f = 24. \nonumber \] Finally, we know that each touchdown was followed by a successful extra point or two point conversion. This is subtle, but it tells us that the number of touchdowns is equal to the sum of extra points and two point conversions. In other words, \[t = x+w. \nonumber \]

To solve our problem, we put these equations into a matrix and put the matrix into reduced row echelon form. Doing so, we find

\[\left[\begin{array}{ccccc}{1}&{1}&{1}&{1}&{7}\\{6}&{1}&{2}&{3}&{24}\\{1}&{-1}&{-1}&{0}&{0}\end{array}\right]\qquad\overrightarrow{\text{rref}}\qquad\left[\begin{array}{ccccc}{1}&{0}&{0}&{0.5}&{3.5}\\{0}&{1}&{0}&{1}&{4}\\{0}&{0}&{1}&{-0.5}&{-0.5}\end{array}\right] \nonumber \]

Therefore, we know that \[\begin{align}\begin{aligned} t &=3.5-0.5f\\ x&=4-f\\ w&=-0.5+0.5f. \\ \end{aligned}\end{align} \nonumber \] We recognize that this means there are “infinite solutions,” but of course most of these will not make sense in the context of a real football game. We must apply some logic to make sense of the situation.

Progressing in no particular order, consider the second equation, \(x = 4-f\) . In order for us to have a positive number of extra points, we must have \(f\leq 4\) . (And of course, we need \(f\geq 0\) , too.) Therefore, right away we know we have a total of only 5 possibilities, where \(f = 0\) , \(1\) , \(2\) , \(3\) or \(4\) .

From the first and third equations, we see that if \(f\) is an even number, then \(t\) and \(w\) will both be fractions (for instance, if \(f=0\) , then \(t = 3.5\) ) which does not make sense. Therefore, we are down to two possible solutions, \(f = 1\) and \(f=3\) .

If \(f=1\) , we have 3 touchdowns, 3 extra points, no two point conversions, and (of course), 1 field goal. (Check to make sure that gives 24 points!) If \(f=3\) , then we 2 touchdowns, 1 extra point, 1 two point conversion, and (of course) 3 field goals. Again, check to make sure this gives us 24 points. Also, we should check each solution to make sure that we have a total of 7 scoring occasions and that each touchdown could be followed by an extra point or a two point conversion.

We have seen a variety of applications of systems of linear equations. We would do well to remind ourselves of the ways in which solutions to linear systems come: there can be exactly one solution, infinite solutions, or no solutions. While we did see a few examples where it seemed like we had only 2 solutions, this was because we were restricting our solutions to “make sense” within a certain context.

We should also remind ourselves that linear equations are immensely important. The examples we considered here ask fundamentally simple questions like “How fast is the water moving?” or “What is the quadratic function that goes through these three points?” or “How were points in a football game scored?” The real “important” situations ask much more difficult questions that often require thousands of equations! (Gauss began the systematic study of solving systems of linear equations while trying to predict the next sighting of a comet; he needed to solve a system of linear equations that had 17 unknowns. Today, this a relatively easy situation to handle with the help of computers, but to do it by hand is a real pain.) Once we understand the fundamentals of solving systems of equations, we can move on to looking at solving bigger systems of equations; this text focuses on getting us to understand the fundamentals.