3 steps problem solving

• 281.412.7766s
• [email protected]
• Learner Dashboard
• GET YOUR CAUSE MAPPING TEMPLATE

• About Cause Mapping®
• What is Root Cause Analysis?
• Cause Mapping® Method
• Cause Mapping® FAQs
• Why ThinkReliability?
• Online Workshops
• Online Short Courses
• On-Demand Training Catalog
• On-Demand Training Subscription
• Company Case Study
• Upcoming Webinars
• Webinar Archives
• Public Workshops
• Private Workshops
• Cause Mapping Certified Facilitator Program
• Our Services
• Facilitation, Consulting, and Coaching
• Root Cause Analysis Program Development
• Work Process Reliability™
• Cause Mapping® Template
• Root Cause Analysis Examples
• Video Library
• Articles and Downloads
• About ThinkReliability
• Client List
• Testimonials

Problem Solving - 3 Basic Steps

Don't complicate it.

Problems can be confusing. Your problem-solving process shouldn’t make them more confusing. With a variety of different tools available, it’s common for people in the same company to use different approaches and different terminology. This makes problem solving problematic. It shouldn’t be.

Some companies use 5Whys , some use fishbone diagrams , and some categorize incidents into generic buckets like " human error " and " procedure not followed ." Some problem-solving methods have six steps, some have eight steps and some have 14 steps. It’s easy to understand how employees get confused.

6-sigma is another widely recognized problem-solving tool. It has five steps with its own acronym, DMAIC: define, measure, analyze, improve and control. The first two steps are for defining and measuring the problem . The third step is the analysis . And the fourth and fifth steps are improve and control, and address solutions .

3 Basic Steps of Problem Solving

As the name suggests, problem solving starts with a problem and ends with solutions. The step in the middle is the analysis. The level of detail within a problem changes based on the magnitude of an issue, but the basic steps of problem solving remain the same regardless of the type of problem:

Step 1. Problem

Step 2. analysis, step 3. solutions.

But these steps are not necessarily what everyone does. Some groups jump directly to solutions after a hasty problem definition. The analysis step is regularly neglected. Individuals and organizations don’t dig into the details that are essential to understand the issue. In the Cause Mapping® method, the point of root cause analysis is to reveal what happened within an incident—to do that digging.

Step 1. Problem

A complete problem definition consists of several different questions:

• What is the problem?
• When did it happen?
• Where did it happen?
• What was the total impact to each of the organization’s overall goals?

These four questions capture what individuals see as a problem, along with the specifics about the setting of the issue (the time and place), and, importantly, the overall consequences to the organization. The traditional approach of writing a problem description as a few sentences doesn’t necessarily capture the information needed for a complete definition. Some organizations see their problem as a single effect, but that doesn’t reflect the nature of an actual issue since different negative outcomes can occur within the same incident. Specific pieces of information are captured within each of the four questions to provide a thorough definition of the problem.

The analysis step provides a clear explanation of an issue by breaking it down into parts. A simple way to organize the details of an incident is to make a timeline . Each piece of the incident in placed in chronological order. A timeline is an effective way to understand what happened and when for an issue.

Ultimately, the objective of problem solving is to turn the negative outcomes defined in step 1 into positive results. To do so, the causes that produced the unwanted outcomes must be identified. These causes provide both the explanation of the issue as well as control points for different solution options. This cause-and-effect approach is the basis of explaining and preventing a problem solving. It’s why cause-and-effect thinking is fundamental for troubleshooting, critical thinking and effective root cause analysis.

Many organizations are under-analyzing their problems because they stop at generic categories like procedure not followed, training less than adequate or management systems . This is a mistake. Learning how to dig a littler further, by asking more Why questions, can reveal significant insight about those chronic problems that people have come to accept as normal operations.

A Cause Map™ diagram provides a way for frontline personnel, technical leads and managers to communicate the details of an issue objectively, accurately and thoroughly. A cause-and-effect analysis can begin as a single, linear path that can be expanded into as much detail as needed to fully understand the issue.

Solutions are specific actions that control specific causes to produce specific outcomes. Both short-term and long-term solutions can be identified from a clear and accurate analysis. It is also important for people to understand that every cause doesn’t need to be solved. Most people believe that 15 causes require 15 solutions. That is not true. Changing just one cause along a causal path breaks that chain of events. Providing solutions on more than one causal path provides additional layers of protection to further reduce the risk of a similar issue occurring in the future.

The Basics of Problem Solving Don't Change

These three steps of problem solving can be applied consistently across an organization from frontline troubleshooters to the executives. First principles should be the foundation of a company’s problem-solving culture. Overlooking these basics erodes critical thinking. Even though the fundamentals of cause-and-effect don’t change, organizations and individuals continue to find special adjectives, algorithms and jargon appealing. Teaching too many tools and using contrived terms such as “true root causal factors” is a symptom of ignoring lean principles. Don’t do that which is unnecessary.

Your problems may be complex, but your problem-solving process should be clear and simple. A scientific approach that objectively explains what happened and why (cause and effect) is sound. It’s the basis for understanding and solving a problem – any problem. It works on the farm, in the power plant, at the manufacturing company and at an airline. It works for the cancer researcher and for the auto mechanic. It also works the same way for safety incidents, production losses and equipment failures. Cause and effect doesn’t change. Just test it.

If you’re interested in seeing one of your problems dissected as a Cause Map diagram, send us an email or call the ThinkReliability office. We’ll arrange a call to step through your issue. You can also learn more about improving the way your organization investigates and prevents problems through one of our upcoming online webinars, short courses or workshops .

Want to learn more? Watch our 28-minute video on problem-solving basics.

Share This Post With A Friend

Similar Posts

Other resources.

• Root Cause Analysis blog
• Patient Safety blog

READ BY - - - - - - - - - -

Other Resources - - - - - - - - - -

Sign Up For Our eNewsletter

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

Overview of the Problem-Solving Mental Process

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

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

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

Frequently Asked Questions

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

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

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

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

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

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

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

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

2. Defining the Problem

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

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

3. Forming a Strategy

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

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

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

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

4. Organizing Information

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

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

5. Allocating Resources

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

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

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

6. Monitoring Progress

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

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

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

7. Evaluating the Results

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

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

A Word From Verywell​

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

Get Advice From The Verywell Mind Podcast

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

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

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

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

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

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

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

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

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

How it works

For Business

Join Mind Tools

Article • 4 min read

The Problem-Solving Process

Looking at the basic problem-solving process to help keep you on the right track.

By the Mind Tools Content Team

Problem-solving is an important part of planning and decision-making. The process has much in common with the decision-making process, and in the case of complex decisions, can form part of the process itself.

We face and solve problems every day, in a variety of guises and of differing complexity. Some, such as the resolution of a serious complaint, require a significant amount of time, thought and investigation. Others, such as a printer running out of paper, are so quickly resolved they barely register as a problem at all.

Despite the everyday occurrence of problems, many people lack confidence when it comes to solving them, and as a result may chose to stay with the status quo rather than tackle the issue. Broken down into steps, however, the problem-solving process is very simple. While there are many tools and techniques available to help us solve problems, the outline process remains the same.

The main stages of problem-solving are outlined below, though not all are required for every problem that needs to be solved.

1. Define the Problem

Clarify the problem before trying to solve it. A common mistake with problem-solving is to react to what the problem appears to be, rather than what it actually is. Write down a simple statement of the problem, and then underline the key words. Be certain there are no hidden assumptions in the key words you have underlined. One way of doing this is to use a synonym to replace the key words. For example, ‘We need to encourage higher productivity ’ might become ‘We need to promote superior output ’ which has a different meaning.

2. Analyze the Problem

Ask yourself, and others, the following questions.

• Where is the problem occurring?
• When is it occurring?
• Why is it happening?

Be careful not to jump to ‘who is causing the problem?’. When stressed and faced with a problem it is all too easy to assign blame. This, however, can cause negative feeling and does not help to solve the problem. As an example, if an employee is underperforming, the root of the problem might lie in a number of areas, such as lack of training, workplace bullying or management style. To assign immediate blame to the employee would not therefore resolve the underlying issue.

Once the answers to the where, when and why have been determined, the following questions should also be asked:

• Where can further information be found?
• Is this information correct, up-to-date and unbiased?
• What does this information mean in terms of the available options?

3. Generate Potential Solutions

When generating potential solutions it can be a good idea to have a mixture of ‘right brain’ and ‘left brain’ thinkers. In other words, some people who think laterally and some who think logically. This provides a balance in terms of generating the widest possible variety of solutions while also being realistic about what can be achieved. There are many tools and techniques which can help produce solutions, including thinking about the problem from a number of different perspectives, and brainstorming, where a team or individual write as many possibilities as they can think of to encourage lateral thinking and generate a broad range of potential solutions.

4. Select Best Solution

When selecting the best solution, consider:

• Is this a long-term solution, or a ‘quick fix’?
• Is the solution achievable in terms of available resources and time?
• Are there any risks associated with the chosen solution?
• Could the solution, in itself, lead to other problems?

This stage in particular demonstrates why problem-solving and decision-making are so closely related.

5. Take Action

In order to implement the chosen solution effectively, consider the following:

• What will the situation look like when the problem is resolved?
• What needs to be done to implement the solution? Are there systems or processes that need to be adjusted?
• What will be the success indicators?
• What are the timescales for the implementation? Does the scale of the problem/implementation require a project plan?
• Who is responsible?

Once the answers to all the above questions are written down, they can form the basis of an action plan.

6. Monitor and Review

One of the most important factors in successful problem-solving is continual observation and feedback. Use the success indicators in the action plan to monitor progress on a regular basis. Is everything as expected? Is everything on schedule? Keep an eye on priorities and timelines to prevent them from slipping.

If the indicators are not being met, or if timescales are slipping, consider what can be done. Was the plan realistic? If so, are sufficient resources being made available? Are these resources targeting the correct part of the plan? Or does the plan need to be amended? Regular review and discussion of the action plan is important so small adjustments can be made on a regular basis to help keep everything on track.

Once all the indicators have been met and the problem has been resolved, consider what steps can now be taken to prevent this type of problem recurring? It may be that the chosen solution already prevents a recurrence, however if an interim or partial solution has been chosen it is important not to lose momentum.

Problems, by their very nature, will not always fit neatly into a structured problem-solving process. This process, therefore, is designed as a framework which can be adapted to individual needs and nature.

Join Mind Tools and get access to exclusive content.

This resource is only available to Mind Tools members.

Already a member? Please Login here

Get 20% off your first year of Mind Tools

Our on-demand e-learning resources let you learn at your own pace, fitting seamlessly into your busy workday. Join today and save with our limited time offer!

Sign-up to our newsletter

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

Subscribe now

Business Skills

Personal Development

Leadership and Management

Member Extras

Most Popular

Newest Releases

Pain Points Podcast - Balancing Work And Kids

Pain Points Podcast - Improving Culture

Mind Tools Store

About Mind Tools Content

Discover something new today

Pain points podcast - what is ai.

Exploring Artificial Intelligence

Pain Points Podcast - How Do I Get Organized?

It's Time to Get Yourself Sorted!

How Emotionally Intelligent Are You?

Boosting Your People Skills

Self-Assessment

What's Your Leadership Style?

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

Recommended for you

Top tips for staying focused.

If You Have Trouble Concentrating These Tips Will Help You Focus

Business Operations and Process Management

Strategy Tools

Customer Service

Business Ethics and Values

Handling Information and Data

Project Management

Knowledge Management

Self-Development and Goal Setting

Time Management

Presentation Skills

Learning Skills

Career Skills

Communication Skills

Negotiation, Persuasion and Influence

Working With Others

Difficult Conversations

Creativity Tools

Self-Management

Work-Life Balance

Stress Management and Wellbeing

Coaching and Mentoring

Change Management

Team Management

Managing Conflict

Delegation and Empowerment

Performance Management

Leadership Skills

Developing Your Team

Talent Management

Problem Solving

Decision Making

Member Podcast

How to Fix Any Problem: The 3 Step Approach

Solving problems isn't about the what, it's about the how..

Posted April 8, 2015

Your son is struggling with fractions, actually close to tears while doing his homework. Your car has been making some awful rumbly sound that has you worried. Your boyfriend is angry with you—he felt you were curt and cold to his mother when you met her last weekend.

Life and problems, we all know the drill. At 8, it’s math. At 20, it’s your beat up old car. At 30, the boyfriend with his nose out of joint. And multiple times a day there's everything else in between. The content is always a moving target—fractions, boyfriend, car—but by having a solid problem-solving process in place, moving through the content becomes a lot easier. We're back to the difference between the what of our lives and the how, and the how is what counts. As the parent, you want to help your son master fractions, but even more, you want him to learn how to not become overwhelmed and discouraged by teaching him how to approach and manage the problems in his life, whatever they may be. And a lot of us adults have the same struggles.

Here’s a simple roadmap for solving everyday problems along with the places it’s easy to get stuck. We’re talking mundane stuff here. We’re not talking about sorting how the next equation for string theory, or how best to arrange your living room furniture—sure that’s partly about problem solving but more about intuition and innate creativity . And even though we're focusing on the everyday, that doesn't mean that they can't feel overwhelming or that they are not difficult or complex. But that said, the basic problem-solving approach doesn't change. Here goes:

1. Define the problem as concretely / specifically as possible. This is about narrowing your—what is it that needs to get fixed? This creates problem partialization—taking big junks of overwhelming misery and breaking them down into smaller, more manageable bites. It also makes it easier to do the next two steps.

The Trap: Too vague and general. "Can’t do factions math" is not a solvable problem. Neither is your "car seems to be breaking down," your boyfriend is "upset," or that you were "curt." Ditto for being lonely , unfulfilled, unhappy, life sucks, or the couples I see who say they can’t communicate. Yes, you may feel that way, but that is the summary statement to a more specific concrete problem. You need to drill down. Be specific. At what point does the trail of fraction concepts for your son break down? Why this problem and not the one before? Rumbly sound—where does it change when you speed up, etc.? Curt—tell me what thought I did or sounded like that gave you that impression. Can’t communicate—you’re talking so you can communicate. Tell me what is exactly happening when you feel like you are not.

The other trap is that your overwhelming feelings have ramped up so far—your son is on the verge of tears—that it makes the drilling down and defining difficult to start. The problem is no longer the factions but anxiety that needs to be fixed. So you hug him or suggest he take a break and go play outside for awhile, or as an adult you do deep breathing, meditation , exercise, drink chamomile tea, or vent to a friend. Once you're back under your threshold, you move forward.

2. Decide what you can do. As the parent, you can walk through the problem with your child or if it is over your head, you can hire a tutor or call the teacher. As the child, you ask the teacher or the smartest kid in the class for help. The car—if you have mechanic skills, you can check it out yourself. If not, take it to a garage. If you don’t have money to fix it, take the bus 'till you can save up the money or see if your dad can lend you the money. Talk to your boyfriend. Apologize for unintentionally hurting his and his mother’s feelings. Offer to talk to her. Find out what specifically bothered him so much. If, as a couple, you feel you don’t communicate, be proactive and initiate conversations about where you both get stuck in conversations and see where they lead.

You get the idea.

The Trap: Rather than focusing on what can and cannot do, you instead, particularly in relationship problems, tie your solution to what you want someone else to do. Rather than having that conversation with your boyfriend you obsess about his need to simply grow up and not be so sensitive and critical. Rather sitting down with your son and walking step-by-step through the math problems, you get mentally hung up on wishing he would try harder and not just whine.

Hitching your problem-solving wagon to someone else changing is a convoluted path to a solution. Sure, you can snap back at your boyfriend for his immaturity or your son about his whining, but it distracts both of you from solving the immediate problem and often only creates another problem. Keep it simple. Your problem, embrace it.

The other trap is that rather than deciding what you can do, you decide to do nothing, to push the problem to the back burner, hope it will go away somehow, miraculously get better. Sometimes deliberately deciding to wait-and-see has merits, especially if you and/or the other is stressed —this is about lowering the anxiety first. Circle back to the fractions tomorrow, realize that you or your boyfriend are under a lot of stress at work and a heavy conversation right now will only make matters worse, and the car noise hasn't gotten worse and you have too much on your plate to this week to tackle it. This is rational decision-making . But simply pushing it way way back is about denial and magical thinking and emotional rather than rational mind. Don't do this.

3. Take action. Once you've zeroed in on the problem, consider action steps. It's time to take action. Do something! Acting and moving forward will help lower your anxiety and help stop it from staring you in the face or perpetually circling around your brain as chronic worry. So get an estimate for the car repair, talk to the teacher, find a YouTube video on fractions, write a note to your boyfriend. The action empowers you.

The Trap: The big trap here is thinking that you think you need to find the right solution that guarantees success before you can act. Unless you do—you believe—you'll wind up making a big mistake. This is the Ready, Aim, Fire approach to problems where you spend a lot of time sitting on the couch or endless hours on the Internet doing research, or forever talking to friends trying to figure out the perfect course before doing anything.

The other more practical approach is based on Ready, Fire, Aim. Do something and then see what happens next, and adjust. This is how a lot of big problems are eventually solved—think Edison and his trying out 1,000 of filaments for his light bulb before finding the best one—the trial and error, the creating the feedback loop that helps you discover what does and doesn't work.

So you try the conversation or leave the note with your boyfriend and see what happens next. You call the teacher, or walk through the factions with your son and see if he can with your support connect the dots. You look for a hole in the exhaust system, get a second estimate on the car while also approaching your dad for a loan and looking up bus routes. Whatever you do, don't endlessly mull, brood, and obsess. Perfectionism gets in the way of problem-solving because it can freeze decisive action needed to break through to a solution.

That’s it. All this moving through is a matter of practice and attitude, sometimes support, and like most things, it gets easier with repetition. So give this a try.

You can’t make a mistake.

Feel free to follow me on Twitter

Bob Taibbi, L.C.S.W., has 49 years of clinical experience. He is the author of 13 books and over 300 articles and provides training nationally and internationally.

• Find a Therapist
• Find a Treatment Center
• Find a Psychiatrist
• Find a Support Group
• Find Teletherapy
• United States
• Brooklyn, NY
• Chicago, IL
• Houston, TX
• Los Angeles, CA
• New York, NY
• Portland, OR
• San Diego, CA
• San Francisco, CA
• Seattle, WA
• Washington, DC
• Asperger's
• Bipolar Disorder
• Chronic Pain
• Eating Disorders
• Passive Aggression
• Personality
• Goal Setting
• Positive Psychology
• Stopping Smoking
• Low Sexual Desire
• Relationships
• Child Development
• Therapy Center NEW
• Diagnosis Dictionary
• Types of Therapy

Understanding what emotional intelligence looks like and the steps needed to improve it could light a path to a more emotionally adept world.

• Coronavirus Disease 2019
• Affective Forecasting
• Neuroscience

• The Art of Effective Problem Solving: A Step-by-Step Guide
• Learn Lean Sigma
• Problem Solving

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Table of Contents

Problem solving methodologies.

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

• Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
• Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
• Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
• Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
• Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
• Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
• Prevent recurrence: Put in place measures to keep the problem from recurring.
• Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

• Determine the issue: Define the issue clearly, including its impact on the customer.
• Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
• Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
• Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

• What exactly is the issue?
• What are the problem’s symptoms or consequences?
• Who or what is impacted by the issue?
• When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

• What causes the problem?
• Why does it continue?
• Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

• Can the solution be implemented within the available resources, time, and budget?
• What are the possible barriers to implementing the solution?
• Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

• Novick, L.R. and Bassok, M., 2005.  Problem Solving . Cambridge University Press.

Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

Free Lean Six Sigma Templates

Improve your Lean Six Sigma projects with our free templates. They're designed to make implementation and management easier, helping you achieve better results.

5S Floor Marking Best Practices

In lean manufacturing, the 5S System is a foundational tool, involving the steps: Sort, Set…

How to Measure the ROI of Continuous Improvement Initiatives

When it comes to business, knowing the value you’re getting for your money is crucial,…

8D Problem-Solving: Common Mistakes to Avoid

In today’s competitive business landscape, effective problem-solving is the cornerstone of organizational success. The 8D…

The Evolution of 8D Problem-Solving: From Basics to Excellence

In a world where efficiency and effectiveness are more than just buzzwords, the need for…

8D: Tools and Techniques

Are you grappling with recurring problems in your organization and searching for a structured way…

How to Select the Right Lean Six Sigma Projects: A Comprehensive Guide

Going on a Lean Six Sigma journey is an invigorating experience filled with opportunities for…

How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

Want to subscribe to The McKinsey Podcast ?

Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

Would you like to learn more about our Strategy & Corporate Finance Practice ?

Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

Want better strategies? Become a bulletproof problem solver

Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

Explore a career with us

Related articles.

Strategy to beat the odds

Five routes to more innovative problem solving

• Top Courses
• Online Degrees
• Find your New Career
• Join for Free

7 Problem-Solving Skills That Can Help You Be a More Successful Manager

Discover what problem-solving is, and why it's important for managers. Understand the steps of the process and learn about seven problem-solving skills.

1Managers oversee the day-to-day operations of a particular department, and sometimes a whole company, using their problem-solving skills regularly. Managers with good problem-solving skills can help ensure companies run smoothly and prosper.

If you're a current manager or are striving to become one, read this guide to discover what problem-solving skills are and why it's important for managers to have them. Learn the steps of the problem-solving process, and explore seven skills that can help make problem-solving easier and more effective.

What is problem-solving?

Problem-solving is both an ability and a process. As an ability, problem-solving can aid in resolving issues faced in different environments like home, school, abroad, and social situations, among others. As a process, problem-solving involves a series of steps for finding solutions to questions or concerns that arise throughout life.

The importance of problem-solving for managers

Managers deal with problems regularly, whether supervising a staff of two or 100. When people solve problems quickly and effectively, workplaces can benefit in a number of ways. These include:

Greater creativity

Higher productivity

Increased job fulfillment

Satisfied clients or customers

Better cooperation and cohesion

Improved environments for employees and customers

7 skills that make problem-solving easier

Companies depend on managers who can solve problems adeptly. Although problem-solving is a skill in its own right, a subset of seven skills can help make the process of problem-solving easier. These include analysis, communication, emotional intelligence, resilience, creativity, adaptability, and teamwork.

1. Analysis

As a manager , you'll solve each problem by assessing the situation first. Then, you’ll use analytical skills to distinguish between ineffective and effective solutions.

2. Communication

Effective communication plays a significant role in problem-solving, particularly when others are involved. Some skills that can help enhance communication at work include active listening, speaking with an even tone and volume, and supporting verbal information with written communication.

3. Emotional intelligence

Emotional intelligence is the ability to recognize and manage emotions in any situation. People with emotional intelligence usually solve problems calmly and systematically, which often yields better results.

4. Resilience

Emotional intelligence and resilience are closely related traits. Resiliency is the ability to cope with and bounce back quickly from difficult situations. Those who possess resilience are often capable of accurately interpreting people and situations, which can be incredibly advantageous when difficulties arise.

5. Creativity 

When brainstorming solutions to problems, creativity can help you to think outside the box. Problem-solving strategies can be enhanced with the application of creative techniques. You can use creativity to:

Approach problems from different angles

Improve your problem-solving process

Spark creativity in your employees and peers

6. Adaptability

Adaptability is the capacity to adjust to change. When a particular solution to an issue doesn't work, an adaptable person can revisit the concern to think up another one without getting frustrated.

7. Teamwork

Finding a solution to a problem regularly involves working in a team. Good teamwork requires being comfortable working with others and collaborating with them, which can result in better problem-solving overall.

Steps of the problem-solving process

Effective problem-solving involves five essential steps. One way to remember them is through the IDEAL model created in 1984 by psychology professors John D. Bransford and Barry S. Stein [ 1 ]. The steps to solving problems in this model include: identifying that there is a problem, defining the goals you hope to achieve, exploring potential solutions, choosing a solution and acting on it, and looking at (or evaluating) the outcome.

1. Identify that there is a problem and root out its cause.

To solve a problem, you must first admit that one exists to then find its root cause. Finding the cause of the problem may involve asking questions like:

Can the problem be solved?

How big of a problem is it?

Why do I think the problem is occurring?

What are some things I know about the situation?

What are some things I don't know about the situation?

Are there any people who contributed to the problem?

Are there materials or processes that contributed to the problem?

Are there any patterns I can identify?

2. Define the goals you hope to achieve.

Every problem is different. The goals you hope to achieve when problem-solving depend on the scope of the problem. Some examples of goals you might set include:

Gather as much factual information as possible.

Brainstorm many different strategies to come up with the best one.

Be flexible when considering other viewpoints.

Articulate clearly and encourage questions, so everyone involved is on the same page.

Be open to other strategies if the chosen strategy doesn't work.

Stay positive throughout the process.

3. Explore potential solutions.

Once you've defined the goals you hope to achieve when problem-solving , it's time to start the process. This involves steps that often include fact-finding, brainstorming, prioritizing solutions, and assessing the cost of top solutions in terms of time, labor, and money.

4. Choose a solution and act on it.

Evaluate the pros and cons of each potential solution, and choose the one most likely to solve the problem within your given budget, abilities, and resources. Once you choose a solution, it's important to make a commitment and see it through. Draw up a plan of action for implementation, and share it with all involved parties clearly and effectively, both verbally and in writing. Make sure everyone understands their role for a successful conclusion.

5. Look at (or evaluate) the outcome.

Evaluation offers insights into your current situation and future problem-solving. When evaluating the outcome, ask yourself questions like:

Did the solution work?

Will this solution work for other problems?

Were there any changes you would have made?

Would another solution have worked better?

As a current or future manager looking to build your problem-solving skills, it is often helpful to take a professional course. Consider Improving Communication Skills offered by the University of Pennsylvania on Coursera. You'll learn how to boost your ability to persuade, ask questions, negotiate, apologize, and more. 

You might also consider taking Emotional Intelligence: Cultivating Immensely Human Interactions , offered by the University of Michigan on Coursera. You'll explore the interpersonal and intrapersonal skills common to people with emotional intelligence, and you'll learn how emotional intelligence is connected to team success and leadership.

Build job-ready skills with a Coursera Plus subscription

• Get access to 7,000+ learning programs from world-class universities and companies, including Google, Yale, Salesforce, and more
• Try different courses and find your best fit at no additional cost
• Earn certificates for learning programs you complete
• A subscription price of $59/month, cancel anytime

Article sources

Tennessee Tech. “ The Ideal Problem Solver (2nd ed.) , https://www.tntech.edu/cat/pdf/useful_links/idealproblemsolver.pdf.” Accessed December 6, 2022.

Keep reading

Coursera staff.

Editorial Team

Coursera’s editorial team is comprised of highly experienced professional editors, writers, and fact...

This content has been made available for informational purposes only. Learners are advised to conduct additional research to ensure that courses and other credentials pursued meet their personal, professional, and financial goals.

35 problem-solving techniques and methods for solving complex problems

Design your next session with SessionLab

Join the 150,000+ facilitators 
using SessionLab.

Recommended Articles

A step-by-step guide to planning a workshop, how to create an unforgettable training session in 8 simple steps, 47 useful online tools for workshop planning and meeting facilitation.

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!

Leave a Comment Cancel reply

Your email address will not be published. Required fields are marked *

Going from a mere idea to a workshop that delivers results for your clients can feel like a daunting task. In this piece, we will shine a light on all the work behind the scenes and help you learn how to plan a workshop from start to finish. On a good day, facilitation can feel like effortless magic, but that is mostly the result of backstage work, foresight, and a lot of careful planning. Read on to learn a step-by-step approach to breaking the process of planning a workshop into small, manageable chunks.  The flow starts with the first meeting with a client to define the purposes of a workshop.…

How does learning work? A clever 9-year-old once told me: “I know I am learning something new when I am surprised.” The science of adult learning tells us that, in order to learn new skills (which, unsurprisingly, is harder for adults to do than kids) grown-ups need to first get into a specific headspace.  In a business, this approach is often employed in a training session where employees learn new skills or work on professional development. But how do you ensure your training is effective? In this guide, we'll explore how to create an effective training session plan and run engaging training sessions. As team leader, project manager, or consultant,…

Effective online tools are a necessity for smooth and engaging virtual workshops and meetings. But how do you choose the right ones? Do you sometimes feel that the good old pen and paper or MS Office toolkit and email leaves you struggling to stay on top of managing and delivering your workshop? Fortunately, there are plenty of online tools to make your life easier when you need to facilitate a meeting and lead workshops. In this post, we’ll share our favorite online tools you can use to make your job as a facilitator easier. In fact, there are plenty of free online workshop tools and meeting facilitation software you can…

Design your next workshop with SessionLab

Join the 150,000 facilitators using SessionLab

Sign up for free

3 Steps to Successfully Solve Any Problem

At its core, problem-solving is simply the process of identifying and addressing a challenge or obstacle that stands in the way of achieving a goal. While many different strategies and techniques can be used to solve problems effectively, three key steps are essential for any successful problem-solving process: gathering information, developing possible solutions, and evaluating potential solutions.

Whether you are facing a workplace challenge, tackling a personal problem, or working to overcome a difficult obstacle, the key to success is assessing the situation thoroughly, gathering all of the necessary information, and carefully evaluate your options.

This may involve brainstorming potential solutions with colleagues or seeking input from experts, as well as carefully considering the possible outcomes of each option. Ultimately, the key is to be persistent and remain focused on finding a solution that works for you. With the right mindset and approach, any problem can be successfully solved.

Are you tired of wasting time on problems that you can't solve? This guide will systematically show you how to solve any problem in just three steps.

Whether you are facing a work challenge, tackling a personal problem, or struggling to overcome an obstacle, the key to success has the right mindset and approach to problem-solving. At its core, problem-solving involves:

Identifying the challenge or obstacle that stands in your way.

Gathering information to understand the situation entirely.

Considering all of your possible solutions before deciding on the best course of action.

To solve a problem, you need to be able to gather enough and the right kind of information.

To successfully solve a problem, you must have accurate information about that problem. This involves being able to gather the right kind of data, as well as having the knowledge and skills needed to interpret it effectively. One essential part of problem-solving is analyzing the problem, which requires you to break it down into smaller components to understand its root causes better.

Once you have gathered enough data and understand the problem, you can begin considering possible solutions and selecting the best ones based on your available resources. Ultimately, analyzing a problem and gathering relevant information is crucial for achieving effective problem-solving results.

Developing possible solutions is an essential step in the problem-solving process steps.

Once you clearly understand the problem, your next step is to start thinking creatively about possible solutions. This may involve brainstorming ideas with colleagues or seeking input from experts, as well as considering potential outcomes for each option and weighing the pros and cons of each solution before making a final decision.

In many cases, trying out different solutions to see what works best can also be helpful. Then, with persistence, focus, and creativity, you can develop practical solutions that will allow you to overcome any challenge or obstacle that stands in your way.

Evaluating potential solutions is key to ensuring that you choose the right approach to solve your problem.

Before making a final decision about the best course of action, it is essential to evaluate your potential solutions and consider the possible outcomes carefully. This may involve conducting research, consulting with subject matter experts, or running tests to determine which solution will most effectively address your challenge or obstacle. It is also essential to consider any risks associated with each option and how feasible it will be to implement the chosen solution based on your available resources.

With a clear understanding of the problem, the right mindset and approach for problem-solving, and a willingness to explore different options, you can successfully overcome any challenge or obstacle that stands in your way. In addition, adopting these essential problem-solving skills will enable you to achieve your goals and live a more fulfilling and successful life at work or in your personal life.

Don't let problems hold you back any longer. With this simple three-step process, you will be able to quickly and effectively solve any problem that comes your way. The first step is gathering enough information about the issue at hand. Once you have a good understanding of what the problem is, you can begin developing possible solutions.

After narrowing down your options, it's essential to evaluate each potential solution to ensure that you choose the best option for solving your problem. Join our course on problem-solving today and learn how to overcome any obstacle life throws your way.

What's your favorite problem-solving technique?

My favorite problem-solving technique is breaking down the problem into smaller parts and then attacking each piece individually. This involves understanding the problem, devising a plan of action, implementing the program, and checking to ensure the solution solves the problem.

This technique works well because it helps you focus on one task at a time and prevents you from getting overwhelmed by the size or complexity of the problem. It also allows you to test different solutions and see which works best. And finally, it helps you ensure that your solution solves the problem.

Do you prefer to work on problems alone or with others?

I prefer to work on problems with others. I think it's essential to have different perspectives when solving problems. Everyone has strengths and weaknesses, so it's helpful to have as many different viewpoints as possible when trying to solve a problem. Plus, working with others can be a lot of fun!

Have you ever had a problem that you couldn't solve? If so, how did you go about finding a solution?

I have had a problem that I couldn't solve. If so, how did you go about it?

First, I would try to narrow the problem as much as possible. Then, I would research the problem and try to find any potential solutions. After that, I would test those potential solutions to see if they worked.

If none of the possible solutions worked, I would start from scratch and develop a new plan of action. Finally, I would execute that plan of action and hope for the best.

Is there a problem-solving method that you're particularly interested in but haven't had the opportunity to try yet?

I'm interested in the problem-solving process itself. The problem-solving process entails breaking a problem into smaller and smaller manageable parts. Then, once the smaller pieces are understood, the solution to the original problem can be found.

This approach is often called "Divide and Conquer." And it's a very effective way of solving problems. The key is to break the problem into manageable chunks and take one step at a time.

Of course, if you try to solve the entire problem simultaneously, you will likely become overwhelmed and frustrated. But taking it one step at a time can slowly but surely work toward a solution.

Do you think there's always a solution to every problem, or are some problems unsolvable?

There is never a simple solution to every problem. Many problems don't have a definitive answer. What is important is how we approach problem-solving.

The first step in any problem-solving process is to identify the problem. This cannot be easy because sometimes we are so close to a situation that we can't see it objectively. However, once the issue is identified, we can look for potential solutions.

Not all solutions are viable, and some may even worsen; therefore, it is a problem. It's essential to evaluate all potential solutions and choose the best one. Sometimes this means trying multiple solutions until one works.

Are there any tricks or tactics to help you solve problems more efficiently?

There's no one-size-fits-all answer to this question, as the best way to solve problems will vary depending on the situation. However, a general process that can be useful for solving many types of issues is illustrated in the diagram below.

The first step is to identify and understand the problem. This may involve identifying the problem's root cause and understanding all the relevant facts and figures. Once you have a good understanding of the problem, you can then begin brainstorming possible solutions. After you have a few potential solutions, evaluating them carefully and selecting the best one is essential. 

What are the three main steps of problem-solving?

Solving Problems Step-by-Step The initial phase in problem-solving involves Identifying and Understanding the Problem. This crucial starting point requires to clearly defining the issue. This step necessitates a thorough analysis of what the actual problem is, its contextual elements, and its potential implications. Following the identification is the Developing Possible Solutions stage. You need to brainstorm various strategies to handle the identified problem in this second step. The emphasis here is on generating a wide array of potential solutions. These strategies must be carefully assessed and selected in order to come up with the most effective solution. After you have identified potential solutions, the final step is Implementing the Chosen Solution. This phase requires action. A decision needs to be made on which solution or combination of solutions will be executed. After that, you must follow through by initiating efforts that will lead to the resolution of the problem. In conclusion, the three main steps of problem-solving include Identifying and Understanding the Problem, Developing Possible Solutions, and Implementing the Chosen Solution. These steps equip individuals with the necessary methodologies to navigate through any issue in a systematic and logical manner.

How does future problem-solving differ from traditional problem-solving approaches?

Proactive Approach of Future Problem-Solving Traditional problem-solving methods mainly focus on resolving issues as they arise. This involves identifying a problem, determining its cause, examining potential strategies, implementing a solution, and assessing its effectiveness. They are more reactive in nature, tackling problems that have already occurred. On the contrary, future problem-solving is more about anticipation. Instead of waiting for problems to occur, it assumes probable issues to arise in the future. It then engages in creating strategies to prevent those problems or mitigate their impact. This proactive approach of preemptively addressing potential problems is a key characteristic of future problem-solving. Use of Foresight in Future Problem-Solving In addition to anticipating problems, future problem-solving often uses foresight techniques, such as forecasting or scenario planning. These methods enable a better understanding of potential future environments and how current decisions might impact them. Hence, future problem-solving is not only about solving problems but also about crafting the future. Systems Thinking in Future Problem-Solving Another aspect that sets future problem-solving apart is the use of systems thinking. Instead of looking at problems in isolation, it sees them as part of a larger system. This approach helps in grasping the big picture and understanding the complex interdependence between various elements. In conclusion, future problem-solving surpasses traditional problem-solving just from being reactive to proactive. It is not only about dealing with present realities but also preparing for prospective issues. It leverages foresight tools and systems thinking to understand and shape the future, making it a more comprehensive and strategic approach to problem-solving.

In the context of future problem-solving, how does one identify potential long-term consequences of a solution?

Identification of Potential Long-Term Consequences In foreseeing the long-term outcomes of a solution, certain strategies can be observed. First and foremost, one must understand the problem comprehensively. By doing so, they position themselves to anticipate the impacts of the solution better. Analyzing Current Trends Analyzing trends associated with the problem helps to predict potential challenges. It involves looking at current patterns within the system and using them to envisage probable impacts. Implementing Scenario Planning Scenario planning avails one with multiple hypothetical situations, giving an array of potential outcomes. It allows decision-makers to examine a diverse range of scenarios and anticipate possible effects. Modeling and Simulation Additionally, the use of modeling and simulation is essential. These tactics offer a visual representation of the likely consequences, making it easier to discern long-term effects. Integration of Diverse Perspectives Involving a diverse group of stakeholders is also helpful. They provide unique insights into potential outcomes, assisting one to perceive the long-term consequences from a more holistic approach. Use of Decision-Making Tools and Techniques Further, one can employ various decision-making tools and techniques. Techniques such as Risk Analysis, SWOT Analysis, and Decision Trees help in predicting long-term consequences, highlighting potential risks and benefits. Continuous Review and Evaluation Finally, a continuous review and evaluation process allows for early identification of the long-term implications. Regular assessments help in detecting unforeseen consequences, aiding in corrective measures. True tailoring of future problem-solving demands imaginative and strategic thinking. Taking steps to identify long-term consequences, as discussed above, is central in developing sustainable solutions. Utilizing these strategies promotes robust, adaptable problem-solving, instrumental in navigating the ever-evolving complexities of the future.

She describes himself as someone who loves to write about digital marketing, social media and public relations. His personal development special interest lies in self-improvement through reading books on the subject of human behavior; she also has an eye for how these topics apply outside just business or career settings too!

Definition of Problem-Solving With Examples

How To Become a Great Problem Solver?

Need a New Problem-Solving Strategy? Try These!

10 Things You Need to Know About Problem Solving

Problem Solving: Tips, Tricks, and Tactics

How To Improve Your Problem-Solving Skills

Which Common Problems Can Be Solved With NLP Techniques?

Problem Solving - Solve any problem in less than 3 minutes

Problem Solving Method Of Teaching

A Problem Solving Method: Brainstorming

What are Problem Solving Skills?

How To Develop Problem Solving Skills?

• Spring 2024
• Spring 2023
• Spring 2022
• Spring 2021
• Spring 2020
• Summer 2019
• Spring 2019
• Summer 2018
• Spring 2018
• Spring 2017
• Spring 2016
• Staff and Contact
• Julia Robinson Math Festival
• Teachers’ Circle
• The Three Stages of the Problem-Solving Cycle

Essentially every problem-solving heuristic in mathematics goes back to George Polya’s How to Solve It ; my approach is no exception. However, this cyclic description might help to keep the process cognitively present.

A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us. During two decades of observing myself and others in the teaching and learning process, I’ve noticed that the most neglected phase is often the first one—understanding the problem.

The Three Stages Explained

• What am I looking for?
• What is the unknown?
• Do I understand every word and concept in the problem?
• Am I familiar with the units in which measurements are given?
• Is there information that seems missing?
• Is there information that seems superfluous?
• Is the source of information bona fide? (Think about those instances when a friend gives you a puzzle to solve and you suspect there’s something wrong with the way the puzzle is posed.)
• Logical reasoning
• Pattern recognition
• Working backwards
• Adopting a different point of view
• Considering extreme cases
• Solving a simpler analogous problem
• Organizing data
• Making a visual representation
• Accounting for all possibilities
• Intelligent guessing and testing

I have produced videos explaining each one of these strategies individually using problems we have solved at the Chapel Hill Math Circle.

• Implementing : We now implement our strategy or set of strategies. As we progress, we check our reasoning and computations (if any). Many novice problem-solvers make the mistake of “doing something” before understanding (or at least thinking they understand) the problem. For instance, if you ask them “What are you looking for?”, they might not be able to answer. Certainly, it is possible to have an incorrect understanding of the problem, but that is different from not even realizing that we have to understand the problem before we attempt to solve it!

As we implement our strategies, we might not be able to solve the problem, but we might refine our understanding of the problem. As we refine our understanding of the problem, we can refine our strategy. As we refine our strategy and implement a new approach, we get closer to solving the problem, and so on. Of course, even after several iterations of this cycle spanning across hours, days, or even years, one may still not be able to solve a particular problem. That’s part of the enchanting beauty of mathematics.

I invite you to observe your own thinking—and that of your students—as you move along the problem-solving cycle!

[1] Problem-Solving Strategies in Mathematics , Posamentier and Krulik, 2015.

About the author: You may contact Hector Rosario at [email protected].

Share this:

1 response to the three stages of the problem-solving cycle.

Pingback: When (Not) to Give a Hint | Chapel Hill Math Circle

Leave a comment Cancel reply

• Search for:

Recent Posts

• When (Not) to Give a Hint
• Making “Maths” Make Sense
• Problem-Solving Pedagogy

Recent Comments

• September 2016
• Uncategorized
• Entries feed
• Comments feed
• WordPress.com

• Already have a WordPress.com account? Log in now.
• Subscribe Subscribed
• Copy shortlink
• Report this content
• View post in Reader
• Manage subscriptions
• Collapse this bar

What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
• How to Resolve Employee Conflict at Work [Steps, Tips, Examples]
• How to Write Inspiring Core Values? 5 Steps with Examples
• 30 Employee Feedback Examples (Positive & Negative)

7.3 Problem-Solving

Learning objectives.

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

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

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

The study of human and animal problem solving processes has provided much insight toward the understanding of our conscious experience and led to advancements in computer science and artificial intelligence. Essentially much of cognitive science today represents studies of how we consciously and unconsciously make decisions and solve problems. For instance, when encountered with a large amount of information, how do we go about making decisions about the most efficient way of sorting and analyzing all the information in order to find what you are looking for as in visual search paradigms in cognitive psychology. Or in a situation where a piece of machinery is not working properly, how do we go about organizing how to address the issue and understand what the cause of the problem might be. How do we sort the procedures that will be needed and focus attention on what is important in order to solve problems efficiently. Within this section we will discuss some of these issues and examine processes related to human, animal and computer problem solving.

PROBLEM-SOLVING STRATEGIES

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

Problems themselves can be classified into two different categories known as ill-defined and well-defined problems (Schacter, 2009). Ill-defined problems represent issues that do not have clear goals, solution paths, or expected solutions whereas well-defined problems have specific goals, clearly defined solutions, and clear expected solutions. Problem solving often incorporates pragmatics (logical reasoning) and semantics (interpretation of meanings behind the problem), and also in many cases require abstract thinking and creativity in order to find novel solutions. Within psychology, problem solving refers to a motivational drive for reading a definite “goal” from a present situation or condition that is either not moving toward that goal, is distant from it, or requires more complex logical analysis for finding a missing description of conditions or steps toward that goal. Processes relating to problem solving include problem finding also known as problem analysis, problem shaping where the organization of the problem occurs, generating alternative strategies, implementation of attempted solutions, and verification of the selected solution. Various methods of studying problem solving exist within the field of psychology including introspection, behavior analysis and behaviorism, simulation, computer modeling, and experimentation.

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

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

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

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

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

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

Further problem solving strategies have been identified (listed below) that incorporate flexible and creative thinking in order to reach solutions efficiently.

Additional Problem Solving Strategies :

• Abstraction – refers to solving the problem within a model of the situation before applying it to reality.
• Analogy – is using a solution that solves a similar problem.
• Brainstorming – refers to collecting an analyzing a large amount of solutions, especially within a group of people, to combine the solutions and developing them until an optimal solution is reached.
• Divide and conquer – breaking down large complex problems into smaller more manageable problems.
• Hypothesis testing – method used in experimentation where an assumption about what would happen in response to manipulating an independent variable is made, and analysis of the affects of the manipulation are made and compared to the original hypothesis.
• Lateral thinking – approaching problems indirectly and creatively by viewing the problem in a new and unusual light.
• Means-ends analysis – choosing and analyzing an action at a series of smaller steps to move closer to the goal.
• Method of focal objects – putting seemingly non-matching characteristics of different procedures together to make something new that will get you closer to the goal.
• Morphological analysis – analyzing the outputs of and interactions of many pieces that together make up a whole system.
• Proof – trying to prove that a problem cannot be solved. Where the proof fails becomes the starting point or solving the problem.
• Reduction – adapting the problem to be as similar problems where a solution exists.
• Research – using existing knowledge or solutions to similar problems to solve the problem.
• Root cause analysis – trying to identify the cause of the problem.

The strategies listed above outline a short summary of methods we use in working toward solutions and also demonstrate how the mind works when being faced with barriers preventing goals to be reached.

One example of means-end analysis can be found by using the Tower of Hanoi paradigm . This paradigm can be modeled as a word problems as demonstrated by the Missionary-Cannibal Problem :

Missionary-Cannibal Problem

Three missionaries and three cannibals are on one side of a river and need to cross to the other side. The only means of crossing is a boat, and the boat can only hold two people at a time. Your goal is to devise a set of moves that will transport all six of the people across the river, being in mind the following constraint: The number of cannibals can never exceed the number of missionaries in any location. Remember that someone will have to also row that boat back across each time.

Hint : At one point in your solution, you will have to send more people back to the original side than you just sent to the destination.

The actual Tower of Hanoi problem consists of three rods sitting vertically on a base with a number of disks of different sizes that can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top making a conical shape. The objective of the puzzle is to move the entire stack to another rod obeying the following rules:

• 1. Only one disk can be moved at a time.
• 2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.
• 3. No disc may be placed on top of a smaller disk.

  Figure 7.02. Steps for solving the Tower of Hanoi in the minimum number of moves when there are 3 disks.

Figure 7.03. Graphical representation of nodes (circles) and moves (lines) of Tower of Hanoi.

The Tower of Hanoi is a frequently used psychological technique to study problem solving and procedure analysis. A variation of the Tower of Hanoi known as the Tower of London has been developed which has been an important tool in the neuropsychological diagnosis of executive function disorders and their treatment.

GESTALT PSYCHOLOGY AND PROBLEM SOLVING

As you may recall from the sensation and perception chapter, Gestalt psychology describes whole patterns, forms and configurations of perception and cognition such as closure, good continuation, and figure-ground. In addition to patterns of perception, Wolfgang Kohler, a German Gestalt psychologist traveled to the Spanish island of Tenerife in order to study animals behavior and problem solving in the anthropoid ape.

As an interesting side note to Kohler’s studies of chimp problem solving, Dr. Ronald Ley, professor of psychology at State University of New York provides evidence in his book A Whisper of Espionage  (1990) suggesting that while collecting data for what would later be his book  The Mentality of Apes (1925) on Tenerife in the Canary Islands between 1914 and 1920, Kohler was additionally an active spy for the German government alerting Germany to ships that were sailing around the Canary Islands. Ley suggests his investigations in England, Germany and elsewhere in Europe confirm that Kohler had served in the German military by building, maintaining and operating a concealed radio that contributed to Germany’s war effort acting as a strategic outpost in the Canary Islands that could monitor naval military activity approaching the north African coast.

While trapped on the island over the course of World War 1, Kohler applied Gestalt principles to animal perception in order to understand how they solve problems. He recognized that the apes on the islands also perceive relations between stimuli and the environment in Gestalt patterns and understand these patterns as wholes as opposed to pieces that make up a whole. Kohler based his theories of animal intelligence on the ability to understand relations between stimuli, and spent much of his time while trapped on the island investigation what he described as  insight , the sudden perception of useful or proper relations. In order to study insight in animals, Kohler would present problems to chimpanzee’s by hanging some banana’s or some kind of food so it was suspended higher than the apes could reach. Within the room, Kohler would arrange a variety of boxes, sticks or other tools the chimpanzees could use by combining in patterns or organizing in a way that would allow them to obtain the food (Kohler & Winter, 1925).

While viewing the chimpanzee’s, Kohler noticed one chimp that was more efficient at solving problems than some of the others. The chimp, named Sultan, was able to use long poles to reach through bars and organize objects in specific patterns to obtain food or other desirables that were originally out of reach. In order to study insight within these chimps, Kohler would remove objects from the room to systematically make the food more difficult to obtain. As the story goes, after removing many of the objects Sultan was used to using to obtain the food, he sat down ad sulked for a while, and then suddenly got up going over to two poles lying on the ground. Without hesitation Sultan put one pole inside the end of the other creating a longer pole that he could use to obtain the food demonstrating an ideal example of what Kohler described as insight. In another situation, Sultan discovered how to stand on a box to reach a banana that was suspended from the rafters illustrating Sultan’s perception of relations and the importance of insight in problem solving.

Grande (another chimp in the group studied by Kohler) builds a three-box structure to reach the bananas, while Sultan watches from the ground.  Insight , sometimes referred to as an “Ah-ha” experience, was the term Kohler used for the sudden perception of useful relations among objects during problem solving (Kohler, 1927; Radvansky & Ashcraft, 2013).

Solving puzzles.

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

How long did it take you to solve this sudoku puzzle? (You can see the answer at the end of this section.)

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

Did you figure it out? (The answer is at the end of this section.) Once you understand how to crack this puzzle, you won’t forget.

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

What steps did you take to solve this puzzle? You can read the solution at the end of this section.

Pitfalls to problem solving.

   Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but she just needs to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

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

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

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

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

Were you able to determine how many marbles are needed to balance the scales in the figure below? You need nine. Were you able to solve the problems in the figures above? Here are the answers.

   Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. Roadblocks to problem solving include a mental set, functional fixedness, and various biases that can cloud decision making skills.

References:

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Review Questions:

1. A specific formula for solving a problem is called ________.

a. an algorithm

b. a heuristic

c. a mental set

d. trial and error

2. Solving the Tower of Hanoi problem tends to utilize a  ________ strategy of problem solving.

a. divide and conquer

b. means-end analysis

d. experiment

3. A mental shortcut in the form of a general problem-solving framework is called ________.

4. Which type of bias involves becoming fixated on a single trait of a problem?

a. anchoring bias

b. confirmation bias

c. representative bias

d. availability bias

5. Which type of bias involves relying on a false stereotype to make a decision?

6. Wolfgang Kohler analyzed behavior of chimpanzees by applying Gestalt principles to describe ________.

a. social adjustment

b. student load payment options

c. emotional learning

d. insight learning

7. ________ is a type of mental set where you cannot perceive an object being used for something other than what it was designed for.

a. functional fixedness

c. working memory

Critical Thinking Questions:

1. What is functional fixedness and how can overcoming it help you solve problems?

2. How does an algorithm save you time and energy when solving a problem?

Personal Application Question:

1. Which type of bias do you recognize in your own decision making processes? How has this bias affected how you’ve made decisions in the past and how can you use your awareness of it to improve your decisions making skills in the future?

anchoring bias

availability heuristic

confirmation bias

functional fixedness

hindsight bias

problem-solving strategy

representative bias

trial and error

working backwards

Answers to Exercises

algorithm:  problem-solving strategy characterized by a specific set of instructions

anchoring bias:  faulty heuristic in which you fixate on a single aspect of a problem to find a solution

availability heuristic:  faulty heuristic in which you make a decision based on information readily available to you

confirmation bias:  faulty heuristic in which you focus on information that confirms your beliefs

functional fixedness:  inability to see an object as useful for any other use other than the one for which it was intended

heuristic:  mental shortcut that saves time when solving a problem

hindsight bias:  belief that the event just experienced was predictable, even though it really wasn’t

mental set:  continually using an old solution to a problem without results

problem-solving strategy:  method for solving problems

representative bias:  faulty heuristic in which you stereotype someone or something without a valid basis for your judgment

trial and error:  problem-solving strategy in which multiple solutions are attempted until the correct one is found

working backwards:  heuristic in which you begin to solve a problem by focusing on the end result

Share This Book

• Increase Font Size

Table of Contents

The problem-solving process, how to solve problems: 5 steps, train to solve problems with lean today, what is problem solving steps, techniques, & best practices explained.

Problem solving is the art of identifying problems and implementing the best possible solutions. Revisiting your problem-solving skills may be the missing piece to leveraging the performance of your business, achieving Lean success, or unlocking your professional potential. 

Ask any colleague if they’re an effective problem-solver and their likely answer will be, “Of course! I solve problems every day.” 

Problem solving is part of most job descriptions, sure. But not everyone can do it consistently. 

Problem solving is the process of defining a problem, identifying its root cause, prioritizing and selecting potential solutions, and implementing the chosen solution.

There’s no one-size-fits-all problem-solving process. Often, it’s a unique methodology that aligns your short- and long-term objectives with the resources at your disposal. Nonetheless, many paradigms center problem solving as a pathway for achieving one’s goals faster and smarter. 

One example is the Six Sigma framework , which emphasizes eliminating errors and refining the customer experience, thereby improving business outcomes. Developed originally by Motorola, the Six Sigma process identifies problems from the perspective of customer satisfaction and improving product delivery. 

Lean management, a similar method, is about streamlining company processes over time so they become “leaner” while producing better outcomes. 

Trendy business management lingo aside, both of these frameworks teach us that investing in your problem solving process for personal and professional arenas will bring better productivity.

1. Precisely Identify Problems

As obvious as it seems, identifying the problem is the first step in the problem-solving process. Pinpointing a problem at the beginning of the process will guide your research, collaboration, and solutions in the right direction. 

At this stage, your task is to identify the scope and substance of the problem. Ask yourself a series of questions: 

• What’s the problem? 
• How many subsets of issues are underneath this problem? 
• What subject areas, departments of work, or functions of business can best define this problem? 

Although some problems are naturally large in scope, precision is key. Write out the problems as statements in planning sheets . Should information or feedback during a later step alter the scope of your problem, revise the statements. 

Framing the problem at this stage will help you stay focused if distractions come up in later stages. Furthermore, how you frame a problem will aid your search for a solution. A strategy of building Lean success, for instance, will emphasize identifying and improving upon inefficient systems. 

2. Collect Information and Plan 

The second step is to collect information and plan the brainstorming process. This is another foundational step to road mapping your problem-solving process. Data, after all, is useful in identifying the scope and substance of your problems. 

Collecting information on the exact details of the problem, however, is done to narrow the brainstorming portion to help you evaluate the outcomes later. Don’t overwhelm yourself with unnecessary information — use the problem statements that you identified in step one as a north star in your research process. 

This stage should also include some planning. Ask yourself:

• What parties will ultimately decide a solution? 
• Whose voices and ideas should be heard in the brainstorming process? 
• What resources are at your disposal for implementing a solution? 

Establish a plan and timeline for steps 3-5. 

3. Brainstorm Solutions

Brainstorming solutions is the bread and butter of the problem-solving process. At this stage, focus on generating creative ideas. As long as the solution directly addresses the problem statements and achieves your goals, don’t immediately rule it out. 

Moreover, solutions are rarely a one-step answer and are more like a roadmap with a set of actions. As you brainstorm ideas, map out these solutions visually and include any relevant factors such as costs involved, action steps, and involved parties. 

With Lean success in mind, stay focused on solutions that minimize waste and improve the flow of business ecosystems. 

Become a Quality Management Professional

• 10% Growth In Jobs Of Quality Managers Profiles By 2025
• 11% Revenue Growth For Organisations Improving Quality

Certified Lean Six Sigma Green Belt

• 4 hands-on projects to perfect the skills learnt
• 4 simulation test papers for self-assessment

Lean Six Sigma Expert

• IASSC® Lean Six Sigma Green Belt and Black Belt certification
• 13 Projects, 12 Simulation exams, 18 Case Studies & 114 PDUs

Here's what learners are saying regarding our programs:

Xueting Liu

Mechanical engineer student at sargents pty. ltd. ,.

A great training and proper exercise with step-by-step guide! I'll give a rating of 10 out of 10 for this training.

Abdus Salam

I have completed the Lean Six Sigma Expert Master’s Program from Simplilearn. And after the course, I could take up new projects and perform better. My average pay rate for a research position increased by 21%.

4. Decide and Implement

The most critical stage is selecting a solution. Easier said than done. Consider the criteria that has arisen in previous steps as you decide on a solution that meets your needs. 

Once you select a course of action, implement it. 

Practicing due diligence in earlier stages of the process will ensure that your chosen course of action has been evaluated from all angles. Often, efficient implementation requires us to act correctly and successfully the first time, rather than being hurried and sloppy. Further compilations will create more problems, bringing you back to step 1. 

5. Evaluate

Exercise humility and evaluate your solution honestly. Did you achieve the results you hoped for? What would you do differently next time? 

As some experts note, formulating feedback channels into your evaluation helps solidify future success. A framework like Lean success, for example, will use certain key performance indicators (KPIs) like quality, delivery success, reducing errors, and more. Establish metrics aligned with company goals to assess your solutions.

Master skills like measurement system analysis, lean principles, hypothesis testing, process analysis and DFSS tools with our Lean Six Sigma Green Belt Training Course . Sign-up today!

Become a quality expert with Simplilearn’s Lean Six Sigma Green Belt . This Lean Six Sigma certification program will help you gain key skills to excel in digital transformation projects while improving quality and ultimate business results.

In this course, you will learn about two critical operations management methodologies – Lean practices and Six Sigma to accelerate business improvement.

Our Quality Management Courses Duration And Fees

Explore our top Quality Management Courses and take the first step towards career success

Get Free Certifications with free video courses

Quality Management

Lean Management

Project Management

Learn from industry experts with free masterclasses, digital marketing.

Digital Marketing Job Search in 2024: Top Things to Do Before You Begin

Unlock Digital Marketing Career Success Secrets for 2024 with Purdue University

Your Gateway to Game-changing Digital Marketing Careers in 2024 with Purdue University

Recommended Reads

Introduction to Machine Learning: A Beginner's Guide

Webinar Wrap-up: Mastering Problem Solving: Career Tips for Digital Transformation Jobs

An Ultimate Guide That Helps You to Develop and Improve Problem Solving in Programming

Free eBook: 21 Resources to Find the Data You Need

ITIL Problem Workaround – A Leader’s Guide to Manage Problems

Your One-Stop Solution to Understand Coin Change Problem

Get Affiliated Certifications with Live Class programs

• PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc.

7.3 Problem Solving

Learning objectives.

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

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

Problem-Solving Strategies

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

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

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

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

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

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

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

Everyday Connection

Solving puzzles.

Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below ( Figure 7.8 ) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

Here is another popular type of puzzle ( Figure 7.9 ) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Take a look at the “Puzzling Scales” logic puzzle below ( Figure 7.10 ). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

Pitfalls to Problem Solving

Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but she just needs to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

Link to Learning

Check out this Apollo 13 scene where the group of NASA engineers are given the task of overcoming functional fixedness.

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

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

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

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

Please visit this site to see a clever music video that a high school teacher made to explain these and other cognitive biases to his AP psychology students.

Were you able to determine how many marbles are needed to balance the scales in Figure 7.10 ? You need nine. Were you able to solve the problems in Figure 7.8 and Figure 7.9 ? Here are the answers ( Figure 7.11 ).

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/psychology/pages/1-introduction

• Authors: Rose M. Spielman, Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett, Marion Perlmutter
• Publisher/website: OpenStax
• Book title: Psychology
• Publication date: Dec 8, 2014
• Location: Houston, Texas
• Book URL: https://openstax.org/books/psychology/pages/1-introduction
• Section URL: https://openstax.org/books/psychology/pages/7-3-problem-solving

© Feb 9, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Problem Solving Steps: 3 Steps to Take Before You Take Any Steps

Is the first problem solving step really ‘Identify the Problem?’

If you’ve ever watched someone who is tired or emotional struggling to solve a problem, you know that somebody forgot a step in the problem solving process.

People can get so wrapped up in the problem of having a problem that they can’t think straight, they get tunnel vision, and any attempt to start a process will be a waste of time.  But if you begin with these three steps, you are more likely to find good solutions, and emerge better from the experience when it’s all over.

1.  Embrace the Suck

Often our first response when we encounter a problem is to see it as a bad thing, an obstruction to forward movement.  But instead of thinking of it as traffic jam on the path to progress, think of it as a sign that you might be doing something right.

In his book It Worked for Me , Colin Powell even reminds us that having problems is actually a good thing, especially as leaders:

Leadership is solving problems. The day Soldiers stop bringing you their problems is the day you have stopped leading them. – Colin Powell Click To Tweet

If your team has concluded that either you can’t help them or you don’t care, that’s a far worse problem to have.  So focus on the upside of having problems:  somebody trusts you enough and believes you have the necessary problem solving skills to do something about them.

Ok, so in a way, you want to have problems.  Does that mean you have to like them?  Of course not.

During my time in the Army, we would often coach ourselves to “ Embrace the Suck .”

Some stuff will come along that you are just going to have to deal with.  You will have problems to solve.  The thing to do is to accept that fact, and get on with the business of solving them.

Two ways to help you past this point are to breathe deeply, and make yourself smile.  These simple acts will help you  keep your head in a crisis  and prepare your mind to think more rationally.

If it helps, tell yourself to “Embrace the Suck.”

2. Distance Yourself

In Scientific American , Oren Shapira and Nira Liberman talk about “Construal Level Theory” – the idea that by increasing our mental distance from a problem we are able to think more abstractly about it.

They cited experiments in which two groups of people were given a transportation problem to help solve.  One was told that it concerned a group of fellow students locally on campus; the other was told it concerned students in a distant country.

Researchers found that the group with the more distant problem produced more possible solutions than the close-to-home group.  Further, their solutions tended to be more creative.  The sense of distance made the problem more abstract and easier to manipulate in their minds.

As the authors concluded, “Even minimal cues of psychological distance can make us more creative.”  Here are four ways you can build some of that helpful mental distance:

Change Perspective – What if it was someone else’s problem?  How would you counsel them to proceed?

Alter Time – What if you had three weeks instead of three days?  Or, if there’s time, consider sleeping on it – things might look different in the morning.

Reduce Probability –  What if it were less likely the problem was going to materialize?  With the pressure off, what might you do differently?

Assign Identity – This wasn’t in the study, but I think another way to separate yourself from the problem is to give it a name.

Something with its own name has its own identity, and is more clearly defined.  That allows you to put some distance between yourself and it.

So give the problem a name.  You can be descriptive, with names like “ The Shipping Snafu ” or the “ The Packaging Predicament. ”  Or have a little fun with it and give it a pet name, like “ Fifi ,” or “ Barney .”

Even the World Meteorological Organization does this when it names storms.  It helps them define what they are talking about and reduce the potential for confusion.

Putting a name on a problem doesn’t make it any less difficult.  But it does help you separate yourself from the problem, and gives you a handle with which you can manipulate it creatively in the abstract to find better solutions.

3. See the Opportunity

The third thing to do is to see the problem as an opportunity for growth.  Assuming you can’t avoid the problem, view it as a chance to sharpen your problem solving skills.

A smooth sea never made a skilled sailor. – F.D. Roosevelt Click To Tweet

The effort of grappling with problems will make you more adept at solving them.  Like your daily workout or playing your ukulele, you get better at it with practice.

It works the same way with your mind, too – it’s like a muscle that grows stronger under manageable levels of stress.  You build mental toughness one day and one challenge at a time by dealing with the problem, not avoiding it.

Viktor Frankl, who has seen more than his share of problems and suffering in Nazi concentration camps, sees it the same way.

What man actually needs is not a tensionless state but rather the striving and struggling for some goal worthy of him. - Viktor Frankl Click To Tweet

I think he’s telling us that we reach our full potential as humans and leaders in the struggle.  So the thing to do is to see the problem as an opportunity to become a better version of yourself and a chance to make things better for others.

Leaders see the struggle as an opportunity to become better versions of themselves while making things better for others. Click To Tweet

Accept the challenge.

Problem Solving Steps – The Takeaway

Problems are a given; you will encounter them (at least I hope you do!).  How you approach the problems you face can have a huge impact on whether or not you find good solutions.

So before you launch into the first problem solving step:   Identify the Problem , take a moment to make sure you are approaching it with the right attitude:

When an unavoidable problem comes running up to you:

Smile and give it a hug.

Name it “Snoopy.”

And take it out for a walk.

The exercise will do you good.

Like what you see? Raise your leadership game. Get access to more tips, tools, and techniques. Subscribe, and I’ll send them direct to your inbox twice a month.

It’s free, I won’t give away your address, and you can unsubscribe any time.

Ken served for 26 years in the Infantry, retiring as a Colonel.  From leading patrols in the Korean DMZ, to parachuting into the jungles of Panama, to commanding a remote outpost on the Iran-Iraq border, he has learned a lot about leadership, and has a passion for sharing that knowledge with others.  Look for his weekly posts, check out his online courses , subscribe below, or simply connect , he loves to talk about this stuff.

GET LEADERSHIP TOOLS

RAPIDSTART LEADERSHIP © 2015-2023   Ι   ALL RIGHTS RESERVED

They are ready to follow...

...are you ready to lead, subscribe now.

Master the 7-Step Problem-Solving Process for Better Decision-Making

Discover the powerful 7-Step Problem-Solving Process to make better decisions and achieve better outcomes. Master the art of problem-solving in this comprehensive guide. Download the Free PowerPoint and PDF Template.

StrategyPunk

Introduction

Mastering the art of problem-solving is crucial for making better decisions. Whether you're a student, a business owner, or an employee, problem-solving skills can help you tackle complex issues and find practical solutions. The 7-Step Problem-Solving Process is a proven method that can help you approach problems systematically and efficiently.

The 7-Step Problem-Solving Process involves steps that guide you through the problem-solving process. The first step is to define the problem, followed by disaggregating the problem into smaller, more manageable parts. Next, you prioritize the features and create a work plan to address each. Then, you analyze each piece, synthesize the information, and communicate your findings to others.

By following this process, you can avoid jumping to conclusions, overlooking important details, or making hasty decisions. Instead, you can approach problems with a clear and structured mindset, which can help you make better decisions and achieve better outcomes.

In this article, we'll explore each step of the 7-Step Problem-Solving Process in detail so you can start mastering this valuable skill. You can download the process's free PowerPoint and PDF templates at the end of the blog post .

Step 1: Define the Problem

The first step in the problem-solving process is to define the problem. This step is crucial because finding a solution is only accessible if the problem is clearly defined. The problem must be specific, measurable, and achievable.

One way to define the problem is to ask the right questions. Questions like "What is the problem?" and "What are the causes of the problem?" can help. Gathering data and information about the issue to assist in the definition process is also essential.

Another critical aspect of defining the problem is identifying the stakeholders. Who is affected by it? Who has a stake in finding a solution? Identifying the stakeholders can help ensure that the problem is defined in a way that considers the needs and concerns of all those affected by it.

Once the problem is defined, it is essential to communicate the definition to all stakeholders. This helps to ensure that everyone is on the same page and that there is a shared understanding of the problem.

Step 2: Disaggregate

After defining the problem, the next step in the 7-step problem-solving process is to disaggregate the problem into smaller, more manageable parts. Disaggregation helps break down the problem into smaller pieces that can be analyzed individually. This step is crucial in understanding the root cause of the problem and identifying the most effective solutions.

Disaggregation can be achieved by breaking down the problem into sub-problems, identifying the contributing factors, and analyzing the relationships between these factors. This step helps identify the most critical factors that must be addressed to solve the problem.

A tree or fishbone diagram is one effective way to disaggregate a problem. These diagrams help identify the different factors contributing to the problem and how they are related. Another way is to use a table to list the other factors contributing to the situation and their corresponding impact on the issue.

Disaggregation helps in breaking down complex problems into smaller, more manageable parts. It helps understand the relationships between different factors contributing to the problem and identify the most critical factors that must be addressed. By disaggregating the problem, decision-makers can focus on the most vital areas, leading to more effective solutions.

Step 3: Prioritize

After defining the problem and disaggregating it into smaller parts, the next step in the 7-step problem-solving process is prioritizing the issues that need addressing. Prioritizing helps to focus on the most pressing issues and allocate resources more effectively.

There are several ways to prioritize issues, including:

• Urgency: Prioritize issues based on their urgency. Problems that require immediate attention should be addressed first.
• Impact: Prioritize issues based on their impact on the organization or stakeholders. Problems with a high impact should be given priority.
• Resources: Prioritize issues based on the resources required to address them. Problems that require fewer resources should be dealt with first.

Considering their concerns and needs, it is important to involve stakeholders in the prioritization process. This can be done through surveys, focus groups, or other forms of engagement.

Once the issues have been prioritized, developing a plan of action to address them is essential. This involves identifying the resources required, setting timelines, and assigning responsibilities.

Prioritizing issues is a critical step in problem-solving. By focusing on the most pressing problems, organizations can allocate resources more effectively and make better decisions.

Step 4: Workplan

After defining the problem, disaggregating, and prioritizing the issues, the next step in the 7-step problem-solving process is to develop a work plan. This step involves creating a roadmap that outlines the steps needed to solve the problem.

The work plan should include a list of tasks, deadlines, and responsibilities for each team member involved in the problem-solving process. Assigning tasks based on each team member's strengths and expertise ensures the work is completed efficiently and effectively.

Creating a work plan can help keep the team on track and ensure everyone is working towards the same goal. It can also help to identify potential roadblocks or challenges that may arise during the problem-solving process and develop contingency plans to address them.

Several tools and techniques can be used to develop a work plan, including Gantt charts, flowcharts, and mind maps. These tools can help to visualize the steps needed to solve the problem and identify dependencies between tasks.

Developing a work plan is a critical step in the problem-solving process. It provides a clear roadmap for solving the problem and ensures everyone involved is aligned and working towards the same goal.

Step 5: Analysis

Once the problem has been defined and disaggregated, the next step is to analyze the information gathered. This step involves examining the data, identifying patterns, and determining the root cause of the problem.

Several methods can be used during the analysis phase, including:

• Root cause analysis
• Pareto analysis
• SWOT analysis

Root cause analysis is a popular method used to identify the underlying cause of a problem. This method involves asking a series of "why" questions to get to the root cause of the issue.

Pareto analysis is another method that can be used during the analysis phase. This method involves identifying the 20% of causes responsible for 80% of the problems. By focusing on these critical causes, organizations can make significant improvements.

Finally, SWOT analysis is a valuable tool for analyzing the internal and external factors that may impact the problem. This method involves identifying the strengths, weaknesses, opportunities, and threats related to the issue.

Overall, the analysis phase is critical for identifying the root cause of the problem and developing practical solutions. Organizations can gain a deeper understanding of the issue and make informed decisions by using a combination of methods.

Step 6: Synthesize

Once the analysis phase is complete, it is time to synthesize the information gathered to arrive at a solution. During this step, the focus is on identifying the most viable solution that addresses the problem. This involves examining and combining the analysis results for a clear and concise conclusion.

One way to synthesize the information is to use a decision matrix. This involves creating a table that lists the potential solutions and the essential criteria for making a decision. Each answer is then rated against each standard, and the scores are tallied to arrive at a final decision.

Another approach to synthesizing the information is to use a mind map. This involves creating a visual representation of the problem and the potential solutions. The mind map can identify the relationships between the different pieces of information and help prioritize the solutions.

During the synthesis phase, remaining open-minded and considering all potential solutions is vital. To ensure everyone's perspectives are considered, it is also essential to involve all stakeholders in the decision-making process.

Step 7: Communicate

After synthesizing the information, the next step is communicating the findings to the relevant stakeholders. This is a crucial step because it helps to ensure that everyone is on the same page and that the decision-making process is transparent.

One effective way to communicate the findings is through a well-organized report. The report should include the problem statement, the analysis, the synthesis, and the recommended solution. It should be clear, concise, and easy to understand.

In addition to the report, a presentation explaining the findings is essential. The presentation should be tailored to the audience and highlight the report's key points. Visual aids such as tables, graphs, and charts can make the presentation more engaging.

During the presentation, it is essential to be open to feedback and questions from the audience. This helps ensure everyone agrees with the recommended solution and addresses concerns or objections.

Effective communication is vital to ensuring the decision-making process is successful. Stakeholders can make informed decisions and work towards a common goal by communicating the findings clearly and concisely.

The 7-step problem-solving process is a powerful tool for helping individuals and organizations make better decisions. By following these steps, individuals can identify the root cause of a problem, prioritize potential solutions, and develop a clear plan of action. This process can be applied to various scenarios, from personal challenges to complex business problems.

Through disaggregation, individuals can break down complex problems into smaller, more manageable parts. By prioritizing potential solutions, individuals can focus their efforts on the most impactful actions. The work step allows individuals to develop a clear action plan, while the analysis step provides a framework for evaluating possible solutions.

The synthesis step combines all the information gathered to develop a comprehensive solution. Finally, the communication step allows individuals to share their answers with others and gather feedback.

By mastering the 7-step problem-solving process, individuals can become more effective decision-makers and problem-solvers. This process can help individuals and organizations save time and resources while improving outcomes. With practice, individuals can develop the skills to apply this process to a wide range of scenarios and make better decisions in all areas of life.

7-Step Problem-Solving Process PPT Template

Free powerpoint and pdf template, executive summary: the 7-step problem-solving process.

The 7-Step Problem-Solving Process is a robust and systematic method to help individuals and organizations make better decisions by tackling complex issues and finding practical solutions. This process comprises defining the problem, disaggregating it into smaller parts, prioritizing the issues, creating a work plan, analyzing the data, synthesizing the information, and communicating the findings.

By following these steps, individuals can identify the root cause of a problem, break it down into manageable components, and prioritize the most impactful actions. The work plan, analysis, and synthesis steps provide a framework for developing comprehensive solutions, while the communication step ensures transparency and stakeholder engagement.

Mastering this process can improve decision-making and problem-solving capabilities, save time and resources, and improve outcomes in personal and professional contexts.

Please buy me a coffee.

I'd appreciate your support if my templates have saved you time or helped you start a project. Buy Me a Coffee is a simple way to show your appreciation and help me continue creating high-quality templates that meet your needs.

7-Step Problem-Solving Process PDF Template

7-step problem-solving process powerpoint template.

Xpeng SWOT Analysis: Free PPT Template and In-Depth Insights (free file)

Unlock key insights into Xpeng with our free SWOT analysis PPT template. Dive deep into its business dynamics at no cost.

Strategic Insights 2024: A SWOT Analysis of Nestle (Plus Free PPT)

Explore Nestle's strategic outlook with our SWOT analysis for 2024. This PowerPoint template highlights key areas for growth and challenges.

2024 Business Disruption: Navigating Growth Through Shaping Strategy

Discover the importance of being a shaper in 2023's business ecosystem. Shaping strategy, attracting a critical mass of participants, and finding the right strategic path to create value.

Solver Title

Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• x^{2}-x-6=0
• -x+3\gt 2x+1
• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• Practice Makes Perfect Learning math takes practice, lots of practice. Just like running, it takes practice and dedication. If you want...

Please add a message.

Message received. Thanks for the feedback.

• Our Mission

3 Simple Strategies to Improve Students’ Problem-Solving Skills

These strategies are designed to make sure students have a good understanding of problems before attempting to solve them.

Research provides a striking revelation about problem solvers. The best problem solvers approach problems much differently than novices. For instance, one meta-study showed that when experts evaluate graphs , they tend to spend less time on tasks and answer choices and more time on evaluating the axes’ labels and the relationships of variables within the graphs. In other words, they spend more time up front making sense of the data before moving to addressing the task.

While slower in solving problems, experts use this additional up-front time to more efficiently and effectively solve the problem. In one study, researchers found that experts were much better at “information extraction” or pulling the information they needed to solve the problem later in the problem than novices. This was due to the fact that they started a problem-solving process by evaluating specific assumptions within problems, asking predictive questions, and then comparing and contrasting their predictions with results. For example, expert problem solvers look at the problem context and ask a number of questions:

• What do we know about the context of the problem?
• What assumptions are underlying the problem? What’s the story here?
• What qualitative and quantitative information is pertinent?
• What might the problem context be telling us? What questions arise from the information we are reading or reviewing?
• What are important trends and patterns?

As such, expert problem solvers don’t jump to the presented problem or rush to solutions. They invest the time necessary to make sense of the problem.

Now, think about your own students: Do they immediately jump to the question, or do they take time to understand the problem context? Do they identify the relevant variables, look for patterns, and then focus on the specific tasks?

If your students are struggling to develop the habit of sense-making in a problem- solving context, this is a perfect time to incorporate a few short and sharp strategies to support them.

3 Ways to Improve Student Problem-Solving

1. Slow reveal graphs: The brilliant strategy crafted by K–8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students’ questioning, sense-making, and evaluating predictions.

For instance, in one third-grade class, students are given a bar graph without any labels or identifying information except for bars emerging from a horizontal line on the bottom of the slide. Over time, students learn about the categories on the x -axis (types of animals) and the quantities specified on the y -axis (number of baby teeth).

The graphs and the topics range in complexity from studying the standard deviation of temperatures in Antarctica to the use of scatterplots to compare working hours across OECD (Organization for Economic Cooperation and Development) countries. The website offers a number of graphs on Google Slides and suggests questions that teachers may ask students. Furthermore, this site allows teachers to search by type of graph (e.g., scatterplot) or topic (e.g., social justice).

2. Three reads: The three-reads strategy tasks students with evaluating a word problem in three different ways . First, students encounter a problem without having access to the question—for instance, “There are 20 kangaroos on the grassland. Three hop away.” Students are expected to discuss the context of the problem without emphasizing the quantities. For instance, a student may say, “We know that there are a total amount of kangaroos, and the total shrinks because some kangaroos hop away.”

Next, students discuss the important quantities and what questions may be generated. Finally, students receive and address the actual problem. Here they can both evaluate how close their predicted questions were from the actual questions and solve the actual problem.

To get started, consider using the numberless word problems on educator Brian Bushart’s site . For those teaching high school, consider using your own textbook word problems for this activity. Simply create three slides to present to students that include context (e.g., on the first slide state, “A salesman sold twice as much pears in the afternoon as in the morning”). The second slide would include quantities (e.g., “He sold 360 kilograms of pears”), and the third slide would include the actual question (e.g., “How many kilograms did he sell in the morning and how many in the afternoon?”). One additional suggestion for teams to consider is to have students solve the questions they generated before revealing the actual question.

3. Three-Act Tasks: Originally created by Dan Meyer, three-act tasks follow the three acts of a story . The first act is typically called the “setup,” followed by the “confrontation” and then the “resolution.”

This storyline process can be used in mathematics in which students encounter a contextual problem (e.g., a pool is being filled with soda). Here students work to identify the important aspects of the problem. During the second act, students build knowledge and skill to solve the problem (e.g., they learn how to calculate the volume of particular spaces). Finally, students solve the problem and evaluate their answers (e.g., how close were their calculations to the actual specifications of the pool and the amount of liquid that filled it).

Often, teachers add a fourth act (i.e., “the sequel”), in which students encounter a similar problem but in a different context (e.g., they have to estimate the volume of a lava lamp). There are also a number of elementary examples that have been developed by math teachers including GFletchy , which offers pre-kindergarten to middle school activities including counting squares , peas in a pod , and shark bait .

Students need to learn how to slow down and think through a problem context. The aforementioned strategies are quick ways teachers can begin to support students in developing the habits needed to effectively and efficiently tackle complex problem-solving.

Follow Polygon online:

• Follow Polygon on Facebook
• Follow Polygon on Youtube
• Follow Polygon on Instagram

Site search

• What to Watch
• What to Play
• PlayStation
• All Entertainment
• Dragon’s Dogma 2
• FF7 Rebirth
• Zelda: Tears of the Kingdom
• Baldur’s Gate 3
• Buyer’s Guides
• Galaxy Brains
• All Podcasts

Filed under:

• Entertainment

The 3-body problem is real, and it’s really unsolvable

Oh god don’t make me explain math

Share this story

• Share this on Facebook
• Share this on Reddit
• Share All sharing options

Share All sharing options for: The 3-body problem is real, and it’s really unsolvable

Everybody seems to be talking about 3 Body Problem , the new Netflix series based on Cixin Liu’s Remembrance of Earth’s Past book trilogy . Fewer people are talking about the two series’ namesake: The unsolvable physics problem of the same name.

This makes sense, because it’s confusing . In physics, the three-body problem attempts to find a way to predict the movements of three objects whose gravity interacts with each of the others — like three stars that are close together in space. Sounds simple enough, right? Yet I myself recently pulled up the Wikipedia article on the three-body problem and closed the tab in the same manner that a person might stagger away from a bright light. Apparently the Earth, sun, and moon are a three-body system? Are you telling me we don’t know how the moon moves ? Scientists have published multiple solutions for the three-body problem? Are you telling me Cixin Liu’s books are out of date?

All I’d wanted to know was why the problem was considered unsolvable, and now memories of my one semester of high school physics were swimming before my eyes like so many glowing doom numbers. However, despite my pains, I have readied several ways that we non-physicists can be confident that the three-body problem is, in fact, unsolvable.

Reason 1: This is a special definition of ‘unsolvable’

The three-body problem is extra confusing, because scientists are seemingly constantly finding new solutions to the three-body problem! They just don’t mean a one-solution-for-all solution. Such a formula does exist for a two-body system, and apparently Isaac Newton figured it out in 1687 . But systems with more than two bodies are, according to physicists, too chaotic (i.e., not in the sense of a child’s messy bedroom, but in the sense of “chaos theory”) to be corralled by a single solution.

When physicists say they have a new solution to the three-body problem, they mean that they’ve found a specific solution for three-body systems that have certain theoretical parameters. Don’t ask me to explain those parameters, because they’re all things like “the three masses are collinear at each instant” or “a zero angular momentum solution with three equal masses moving around a figure-eight shape.” But basically: By narrowing the focus of the problem to certain arrangements of three-body systems, physicists have been able to derive formulas that predict the movements of some of them, like in our solar system. The mass of the Earth and the sun create a “ restricted three-body problem ,” where a less-big body (in this case, the moon) moves under the influence of two massive ones (the Earth and the sun).

What physicists mean when they say the three-body problem has no solution is simply that there isn’t a one-formula-fits-all solution to every way that the gravity of three objects might cause those objects to move — which is exactly what Three-Body Problem bases its whole premise on.

Reason 2: 3 Body Problem picked an unsolved three-body system on purpose

Henri Poincaré’s research into a general solution to the three-body problem formed the basis of what would become known as chaos theory (you might know it from its co-starring role in Jurassic Park ). And 3 Body Problem itself isn’t about any old three-body system. It’s specifically about an extremely chaotic three-body system, the exact kind of arrangement of bodies that Poincaré was focused on when he showed that the problem is “unsolvable.”

[ Ed. note: The rest of this section includes some spoilers for 3 Body Problem .]

In both Liu’s books and Netflix’s 3 Body Problem , humanity faces an invasion by aliens (called Trisolarans in the English translation of the books, and San-Ti in the TV series) whose home solar system features three suns in a chaotic three-body relationship. It is a world where, unlike ours, the heavens are fundamentally unpredictable. Periods of icy cold give way to searing heat that give way to swings in gravity that turn into temporary reprieves that can never be trusted. The unpredictable nature of the San-Ti environment is the source of every detail of their physicality, their philosophy, and their desire to claim Earth for their own.

In other words, 3 Body Problem ’s three-body problem is unsolvable because Liu wanted to write a story with an unsolvable three-body system, so he chose one of the three-body systems for which we have not discovered a solution, and might never.

Reason 3: Scientists are still working on the three-body problem

Perhaps the best reason I can give you to believe that the three-body problem is real, and is really unsolvable, is that some scientists published a whole set of new solutions for specific three-body systems very recently .

If physicists are still working on the three-body problem, we can safely assume that it has not been solved. Scientists, after all, are the real experts. And I am definitely not.

The next level of puzzles.

Take a break from your day by playing a puzzle or two! We’ve got SpellTower, Typeshift, crosswords, and more.

Sign up for the newsletter Patch Notes

A weekly roundup of the best things from Polygon

Just one more thing!

Please check your email to find a confirmation email, and follow the steps to confirm your humanity.

Oops. Something went wrong. Please enter a valid email and try again.

Loading comments...

In defense of Stardew Valley’s fantastically frustrating fishing

How to get the true ending of Dragon’s Dogma 2

• Final Fantasy 14 guides, tips, tricks, and walkthroughs

Which Grand Company should you join in FFXIV?

Marvel Snap Spotlight Cache schedule for April 2024

Spoil yourself with three months of Game Pass Ultimate for $10 off

The Callisto Protocol headlines Humble Choice in April, costing $11.99 to own on PC

Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

Related stories

The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

• Main content

Search code, repositories, users, issues, pull requests...

Provide feedback.

We read every piece of feedback, and take your input very seriously.

Saved searches

Use saved searches to filter your results more quickly.

To see all available qualifiers, see our documentation .

• Notifications

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement . We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

[Question] MLC-LLM (nightly-built) cannot be used because it conflicts with the numpy.How to solve it? #2063

bocurry commented Mar 31, 2024

MasterJH5574 commented Mar 31, 2024

Sorry, something went wrong.

0xDEADFED5 commented Apr 1, 2024

Masterjh5574 commented apr 1, 2024.

No branches or pull requests

Self-adaptive Technique with Double Inertial Steps for Inclusion Problem on Hadamard Manifolds

• Open access
• Published: 02 April 2024

Cite this article

You have full access to this open access article

• Hammed Anuoluwapo Abass   ORCID: orcid.org/0000-0002-4236-3278 1 ,
• Olawale Kazeem Oyewole 2 ,
• Kazeem Olalekan Aremu 1 , 3 &
• Lateef Olakunle Jolaoso 1 , 4  

In this article, we investigate monotone and Lipschitz continuous variational inclusion problem in the settings of Hadamard manifolds. We propose a forward–backward method with a self-adaptive technique for solving variational inclusion problem. To increase the rate of convergence of our proposed method, we incorporate our iterative method with double inertial steps and establish a convergence result of our iterative method under some mild conditions. Finally, in order to illustrate the computational effectiveness of our method, some numerical examples are also discussed. The result present in this article is new in this space and extends many related results in the literature.

Avoid common mistakes on your manuscript.

1 Introduction

Let \(\mathcal {K}\) be a nonempty, closed geodesic convex subset of a Hadamard manifold \(\mathbb {P},~ T_{x}\mathbb {P}\) be the tangent space of \(\mathbb {P}\) at \(x \in \mathbb {P}\) and \(T\mathbb {P}\) be the tangent bundle of \(\mathbb {P}\) . The variational inclusion problem (VIP) is to find \(\overline{x} \in \mathbb {P}\) such that

where \(\Phi : \mathcal {K} \rightarrow T \mathbb {P}\) is a single-valued vector field, \(\Psi : \mathcal {K} \rightarrow 2^{T\mathbb {P}}\) is a multi-valued vector field and 0 denotes the zero section of \(T\mathbb {P}\) . We denote the solution set of ( 1 ) by \(\Omega \) . The variational inclusion problem has received much attention due to its various applications in signal processing, image recovery and statistical regression, (see [ 3 , 13 , 44 , 47 , 49 ]). It is known that several optimization problem such as convex optimization problem can be translated into finding a zero of a maximal monotone operator defined on a Hilbert space \(\mathbb {M}\) . The problem of finding a zero of the sum of two (maximal) monotone operators is of fundamental importance in convex optimization and variational analysis (see [ 1 , 19 , 26 , 33 , 43 , 52 ]). For solving VIP ( 1 ), the forward–backward splitting method (FBM) (see [ 13 , 28 , 29 , 48 ]) is usually employed and is defined in the following manner: \(q_1 \in \mathbb {M}\) and

where \(r > 0, ~\Psi : \mathbb {M} \rightarrow 2^{\mathbb {M}}\) is a set-valued operator and \(\Phi : \mathbb {M} \rightarrow \mathbb {M}\) is an operator. In this case, each step of iterates involves only with \(\Phi \) as the forward step and \(\Psi \) as the backward step, but not the sum of operators. The FBM defined in ( 2 ) above requires one of the operators to be inverse strongly monotone. This assumption imposed on one of the operators is very difficult to meet the practical problems. In order to dispense with the condition, many authors have introduced several iterative methods. For instance, Tseng [ 48 ] introduced the following forward–backward–forward method which is a two-step iterative scheme as follows:

where the step size \(\{r_k\}\) can be updated by Armijo linesearch methods. When the mapping \(\Phi \) is Lipschitz continuous and the mapping \(\Psi \) is maximal monotone, ( 3 ) converges weakly to a solution of VIP in the settings of real Hilbert spaces.

In 2019, Shehu [ 41 ] extended Tseng’s splitting method to the settings of real Banach spaces. He proposed the following iterative method for approximating solution of VIP in a 2-uniformly convex Banach space \(\mathbb {E}\) which is also uniformly smooth as follows:

where \(\Phi : \mathbb {E} \rightarrow \mathbb {E}\) is monotone and L -Lipschitz continuous, \(J_{r_k}^{\Psi }=(J+ r_k \Psi )^{-1}J\) is the resolvent of \(\Psi \) and J is the duality mapping from \(\mathbb {E}\) to \(\mathbb {E}^{*}\) ( \(\mathbb {E}^*\) is the dual of \(\mathbb {E}\) ). He obtained a weak convergence result.

In 1964, Polyak [ 38 ] introduced the inertial extrapolation method which is a useful tool for speeding up the rate of convergence of iterative methods. The idea of inertial extrapolation method was inspired by an implicit discretization of a second-order in-time dissipative dynamical system, so-called heavy ball with friction. The heavy ball friction is a simplified version of the differential system describing the motion of a heavy ball that rolls over the graph f and that keep rolling under its own inertia until friction stop it at a critical point of f . This nonlinear oscillation with damping, which is called the "heavy ball with friction" system, has been considered by several authors from the optimization point of view, establishing different convergence results and identifying circumstances under which the rate of convergence is better than the one of the first-order-steepest descent method (see [ 4 , 6 , 38 ]). Alvarez and Attouch [ 5 ] introduced and constructed the heavy-ball method with the proximal point algorithm to solve a problem of maximal monotone operator. They defined their method as follows:

where \(\{\theta _{k}\} \subset [0,1)\) and \(\{r_k\}\) is nondecreasing with \(\sum \limits _{k=1}^{\infty }\theta _k\Vert q_k-q_{k-1}\Vert < \infty .\) They established that the sequence generated by ( 5 ) converges weakly to a zero of the monotone operator \(\Psi \) . In 2003, Moudafi and Oliny [ 32 ] introduced the following inertial proximal point method for finding the zero of the sum of two monotone operators:

They obtained a weak convergence theorem provided that \(r_k < \frac{2}{L}\) with L being the Lipschitz constant of \(\Phi \) and \(\sum \limits _{k=1}^{\infty }\theta _{k}\Vert q_{k}-q_{k-1}\Vert < \infty \) holds. Polyak [ 37 ] explored the potential of enhancing the convergence speed of numerical iteration methods for solving optimization problems by incorporating multistep inertial extrapolation steps. However, it is important to note that [ 37 , 39 ] do not provide an established convergence analysis or rate of convergence for these multi-step inertial methods. Thus, the use of two or more inertial terms could guarantee necessary acceleration (see [ 30 ]). For growing interests in this direction (see [ 1 , 2 , 24 , 51 ]).

Recently, Dong et al. [ 16 ] introduced the double inertial Mann algorithm and proved the convergence of the proposed algorithm under some suitable conditions: the algorithm is given by

where T is a nonexpansive mapping, \(\lambda , \theta \in [0,1]\) and \(\phi \in (0,1).\)

Very recently, Suantai et al. [ 45 ] also considered a double inertial forward–backward algorithm in the settings of real Hilbert spaces.

Extension of concepts and techniques from linear spaces to Riemannian manifolds has some important advantages (see [ 17 , 22 , 40 ]). For instance, some optimization problems with nonconvex objective functions become convex from the Riemannian geometry point of view, and some constrained optimization problems can be regarded as unconstrained ones with an appropriate Riemannian metric. In addition, the study of convex minimization problems and inclusion problems in nonlinear spaces have proved to be very useful in computing medians and means of trees, which are very important in computational phylogenetics, diffusion tensor imaging, consensus algorithms and modeling of airway systems in human lungs and blood vessels (see [ 9 , 10 , 11 ]). Thus, nonlinear spaces are more suitable frameworks for the study of optimization problems from linear to Riemannian manifolds.

Very recently, Khammahawong et al. [ 20 ] proposed the following forward–backward splitting method for solving variational inclusion problem ( 1 ) in the settings of a Hadamard manifold:

and \(\mu > 0.\) They proved that the sequence by their proposed method converges to an element in \(\Omega \) .

Furthermore, it will be crucial to expand the idea of the double inertial method to the Hadamard manifold because of the significance of our space of interest and the importance of the inertial method in dynamical systems.

Motivated by the aforementioned results in linear and nonlinear spaces, we proposed a forward–backward method together with a double step inertial method for solving variational inclusion problem in the settings of a Hadamard manifold. We prove that the sequence generated by our method converges to a solution of VIP ( 1 ) without the prior knowledge of the Lipschitz constant via a self-adaptive technique. In order to fasten the rate of convergence of our proposed method, we introduce a double inertial steps. Lastly, we compare our results with some related results in the literature to show the performance of our method. To the best of our knowledge, no result on double inertial steps have been discussed in the settings of nonlinear spaces. Our result extends and generalizes many related results in the literature.

2 Preliminaries

Let \(\mathbb {P}\) be an m -dimensional manifold, let \(x \in \mathbb {P}\) and let \(T_x \mathbb {P}\) be the tangent space of \(\mathbb {P}\) at \(x \in \mathbb {P}\) . We denote by \(T\mathbb {P}=\bigcup _{x \in \mathbb {P}}T_x \mathbb {P}\) the tangent bundle of \(\mathbb {P}\) . An inner product \(\mathcal {R}\langle \cdot , \cdot \rangle \) is called a Riemannian metric on \(\mathbb {P}\) if \(\langle \cdot ,\cdot \rangle _x:T_x \mathbb {P} \times T_x \mathbb {P} \rightarrow \mathbb {R}\) is an inner product for all \(x \in \mathbb {P}.\) The corresponding norm induced by the inner product \(\mathcal {R}_x \langle \cdot ,\cdot \rangle \) on \(T_x \mathbb {P}\) is denoted by \(\Vert \cdot \Vert _x.\) We will drop the subscript x and adopt \(\Vert \cdot \Vert \) for the corresponding norm induced by the inner product. A differentiable manifold \(\mathbb {P}\) endowed with a Riemannian metric \(\mathcal {R}\langle \cdot ,\cdot \rangle \) is called a Riemannian manifold. In what follows, we denote the Riemannian metric \(\mathcal {R} \langle \cdot ,\cdot \rangle \) by \(\langle \cdot ,\cdot \rangle \) when no confusion arises. Given a piecewise smooth curve \(\gamma :[a,b] \rightarrow \mathbb {P}\) joining x to y (that is, \(\gamma (a)=x\) and \(\gamma (b)=y\) ), we define the length \(l(\gamma )\) of \(\gamma \) by \(l(\gamma ):= \int _{a}^{b}\Vert \gamma ^{\prime }(t)\Vert \textrm{d}t\) . The Riemannian distance d ( x ,  y ) is the minimal length over the set of all such curves joining x to y . The metric topology induced by d coincides with the original topology on \(\mathbb {P}\) . We denote by \(\nabla \) the Levi-Civita connection associated with the Riemannian metric [ 40 ].

Let \(\gamma \) be a smooth curve in \(\mathbb {P}\) . A vector field X along \(\gamma \) is said to be parallel if \(\nabla _{\gamma ^{\prime }}X=\textbf{0}\) , where \(\textbf{0}\) is the zero tangent vector. If \(\gamma ^{\prime }\) itself is parallel along \(\gamma \) , then we say that \(\gamma \) is a geodesic and \(\Vert \gamma ^{\prime }\Vert \) is a constant. If \(\Vert \gamma ^{\prime }\Vert =1\) , then the geodesic \(\gamma \) is said to be normalized. A geodesic joining x to y in \(\mathbb {P}\) is called a minimizing geodesic if its length equals d ( x ,  y ). A Riemannian manifold \(\mathbb {P}\) equipped with a Riemannian distance d is a metric space \((\mathbb {P},d)\) . A Riemannian manifold \(\mathbb {P}\) is said to be complete if for all \(x \in \mathbb {P}\) , all geodesics emanating from x are defined for all \(t \in \mathbb {R}\) . The Hopf–Rinow theorem [ 40 ] posits that if \(\mathbb {P}\) is complete, then any pair of points in \(\mathbb {P}\) can be joined by a minimizing geodesic. Moreover, if \((\mathbb {P},d)\) is a complete metric space, then every bounded and closed subset of \(\mathbb {P}\) is compact. If \(\mathbb {P}\) is a complete Riemannian manifold, then the exponential map \(\exp _{x}: T_x \mathbb {P} \rightarrow \mathbb {P}\) at \(x \in \mathbb {P}\) is defined by

where \(\gamma _v(\cdot ,x)\) is the geodesic starting from x with velocity v (that is, \(\gamma _v(0,x)=x\) and \(\gamma _{v}^{\prime }(0,x)=v\) ). Then, for any t ,  we have \(\exp _{x}tv=\gamma _v(t,x)\) and \(\exp _{x}{} \textbf{0}=\gamma _v(0,x)=x.\) Note that the mapping \(\exp _x\) is differentiable on \(T_x \mathbb {P}\) for every \(x \in \mathbb {P}.\) The exponential map \(\exp _{x}\) has an inverse \(\exp _{x}^{-1}: \mathbb {P} \rightarrow T_x \mathbb {P}.\) For any \(x,y \in \mathbb {P},\) we have \(d(x,y)=\Vert \exp _{y}^{-1}x\Vert =\Vert \exp _{x}^{-1}y\Vert \) (see [ 40 ] for more details). The parallel transport \(\Gamma _{\gamma ,\gamma (b),\gamma (a)}: T_{\gamma (a)}\mathbb {P} \rightarrow T_{\gamma (b)}\mathbb {P}\) on the tangent bundle \(T\mathbb {P}\) along \(\gamma :[a,b] \rightarrow \mathbb {R}\) with respect to \(\nabla \) is defined by

where F is the unique vector field such that \(\nabla _{\gamma ^{\prime }(t)}v=\textbf{0}\) for all \(t\in [a,b]\) and \(F(\gamma (a))=v.\) If \(\gamma \) is a minimizing geodesic joining x to y ,  then we write \(\Gamma _{y,x}\) instead of \(\Gamma _{\gamma ,y,x}.\) Note that for every \(a,b,r,s \in \mathbb {R},\) we have

Also, \(\Gamma _{\gamma (b),\gamma (a)}\) is an isometry from \(T_{\gamma (a)}\mathbb {P}\) to \(T_{\gamma (b)}\mathbb {P},\) that is, the parallel transport preserves the inner product

Below is an example of a Hadamard manifold.

Space 1: Let \(\mathbb {R}_{++}^{m}\) be the product space \(\mathbb {R}_{++}^m:= \{(x_1,x_2, \cdots , x_m): x_i \in \mathbb {R}_{++},~i=1,2, \cdots , m\}\) . Let \(\mathbb {P} = (\mathbb {R}^m_{++}, \langle \cdot ,\cdot \rangle )\) be the m -dimensional Hadamard manifold with the Riemannian metric \(\langle p,q \rangle =p^\textrm{T} q\) and the distance \(d(x,y)=|\ln \frac{x}{y}|=|\ln \sum \limits _{i=1}^{m}\frac{x_i}{y_i}|,\) where \(x,y \in \mathbb {P}\) with \(x=\{x_i\}_{i=1}^{m}\) and \(y=\{y_i\}_{i=1}^{m}\) .

A subset \(\mathcal {K} \subset \mathbb {P}\) is said to be convex if for any two points \(x,y \in \mathcal {K},\) the geodesic \(\gamma \) joining x to y is contained in \(\mathcal {K}.\) That is, if \(\gamma :[a,b] \rightarrow \mathbb {P}\) is a geodesic such that \(x=\gamma (a)\) and \(y=\gamma (b),\) then \(\gamma ((1-t)a+tb) \in \mathcal {K}\) for all \(t \in [0,1].\) A complete simply connected Riemannian manifold of non-positive sectional curvature is called a Hadamard manifold. We denote by \(\mathbb {P}\) a finite dimensional Hadamard manifold. Henceforth, unless otherwise stated, we represent by \(\mathcal {K}\) a nonempty, closed and convex subset of \(\mathbb {P}.\)

Next, let \(\mathcal {H(K)}\) denote the set of all single-valued vector fields \(U:\mathcal {K} \rightarrow T\mathbb {P}\) such that \(U(p) \in T_{p}\mathbb {P}\) , for each \(p \in \mathcal {K}.\) Let \(\mathcal {X(K)}\) denote to the set of all multivalued vector fields \(V: \mathcal {K} \rightarrow 2^{T\mathbb {P}}\) such that \(V(p)\subseteq T_{p}\mathbb {P}\) for each \(p \in \mathcal {K},\) and the denote \(\textrm{Dom}(V)\) the domain of V defined by \(\textrm{Dom}(V)=\{p \in \mathcal {K}: V(p)\ne \varnothing \}.\)

We state some results and definitions which are needed in the next section.

Definition 1

[ 50 ] A vector field \(U \in \mathcal {H(K)}\) is said to be

monotone, if

L -Lipschitz continuous if there exists \(L> 0\) such that

Definition 2

[ 14 ] A vector field \(V \in \mathcal {X(K)}\) is said to be

monotone, if for all \(p, q \in \textrm{Dom}(V)\) ,

maximal monotone if it is monotone and \(\forall ~p \in \mathcal {K}\) and \(u \in T_{p}\mathcal {K},\) the condition

Definition 3

[ 17 ] Let \(\mathcal {K}\) be a nonempty, closed and subset of \(\mathbb {P}\) and \(\{x_n\}\) be a sequence in \(\mathbb {P}\) . Then, \(\{x_n\}\) is said to be Fejèr convergent with respect to \(\mathcal {K}\) if for all \(p \in \mathcal {K}\) and \(n \in \mathbb {N},\)

Definition 4

[ 25 ] Let \(V \in \mathcal {X(K)}\) be a vector field and \(x_0 \in \mathcal {K}.\) Then, V is said to be upper Kuratowski semicontinuous at \(x_0\) if for any sequences \(\{x_n\} \subseteq \mathcal {K}\) and \(\{v_n\} \subset T\mathbb {P}\) with each \(v_n \in V(x_n),\) the relations \(\lim \limits _{n\rightarrow \infty } {v_n}=v_0\) imply that \(v_0 \in V(x_0)\) . Moreover, V is said to be upper Kuratowski semicontinuous on \(\mathcal {K}\) if it is upper Kuratowski semicontinuous for each \(x \in \mathcal {K}.\)

[ 17 ] Let \(\mathcal {K}\) be a nonempty, closed and closed subset of \(\mathbb {P}\) and \(\{x_n\} \subset \mathbb {P}\) be a sequence such that \(\{x_n\}\) be a Fejér convergent with respect to \(\mathcal {K}.\) Then, the following hold:

For every \(p \in \mathcal {K},~ d(x_n, p)\) converges.

  \(\{x_n\}\) is bounded.

Assume that every cluster point of \(\{x_n\}\) belongs to \(\mathcal {K}\) , then \(\{x_n\}\) converges to a point in \(\mathcal {K}\) .

Proposition 1

[ 40 ]. Let \(x \in \mathbb {P}\) . The exponential mapping \(\exp _{x}: T_x \mathbb {P} \rightarrow \mathbb {P}\) is a diffeomorphism. For any two points \(x,y \in \mathbb {P},\) there exists a unique normalized geodesic joining x to y ,  which is given by

A geodesic triangle \(\Delta (p,q,r)\) of a Riemannian manifold \(\mathbb {P}\) is a set containing three points p ,  q ,  r and three minimizing geodesics joining these points.

Proposition 2

[ 40 ]. Let \(\Delta (p,q,r)\) be a geodesic triangle in \(\mathbb {P}\) . Then

Moreover, if \(\theta \) is the angle at p ,  then we have

[ 25 ] If \(x, y \in \mathbb {P}\) and \(v \in T_{y}\mathbb {P},\) then

[ 21 ] Let \(\mathbb {P}\) be a Hadamard manifold and let \(u,v, w \in \mathbb {P}.\) Then,

[ 25 ] Let \(x_0 \in \mathbb {P}\) and \(\{x_n\} \subset \mathbb {P}\) with \(x_n \rightarrow x_0.\) Then, the following assertions hold:

For any \(y \in \mathbb {P},\) we have \(\exp _{x_n}^{-1}y \rightarrow \exp _{x_0}^{-1}x_n\) and \(\exp _{y}^{-1}x_n \rightarrow \exp _{y}^{-1}x_0.\)

If \(v_n \in T_{x_n}\mathbb {P}\) and \(v_n \rightarrow v_0,\) then \(v_0 \in T_{x_0}\mathbb {P}.\)

Given \(u_n,v_n \in T_{x_n}\mathbb {P}\) and \(u_0, v_0 \in T_{x_0}\mathbb {P}\) , if \(u_n \rightarrow u_0,\) then \(\langle u_n, v_n\rangle \rightarrow \langle u_0, v_0\rangle .\)

For any \(u \in T_{x_0}\mathbb {P},\) the function \(F: \mathbb {P} \rightarrow T\mathbb {P},\) defined by \(F(x)=\Gamma _{x, x_0}u\) for each \(x \in \mathbb {P}\) is continuous on \(\mathbb {P}\) .

The next lemma presents the relationship between triangles in \(\mathbb {R}^2\) and geodesic triangles in Riemannian manifolds (see [ 12 ]).

[ 12 ]. Let \(\Delta (x_1,x_2,x_3)\) be a geodesic triangle in \(\mathbb {P}.\) Then, there exists a triangle \(\Delta (\bar{x}_1,\bar{x}_2,\bar{x}_3)\) corresponding to \(\Delta (x_1,x_2,x_3)\) such that \(d(x_i,x_{i+1})=\Vert \bar{x}_{i}-\bar{x}_{i+1}\Vert \) with the indices taken modulo 3. This triangle is unique up to isometries of \(\mathbb {R}^2.\)

The triangle \(\Delta (\bar{x}_1,\bar{x}_2,\bar{x}_3)\) in Lemma 4 is said to be the comparison triangle for \(\Delta (x_1,x_2,x_3) \subset \mathbb {P}.\) The points \(\bar{x}_1,\) \(\bar{x}_2\) and \(\bar{x}_3\) are called comparison points to the points \(x_1,x_2\) and \(x_3\) in \(\mathbb {P}.\)

A function \(h: \mathbb {P} \rightarrow \mathbb {R}\) is said to be geodesic if for any geodesic \(\gamma \in \mathbb {P},\) the composition \(h \circ \gamma : [u,v] \rightarrow \mathbb {R}\) is convex, that is,

[ 25 ] Let \(\Delta (p,q,r)\) be a geodesic triangle in a Hadamard manifold \(\mathbb {P}\) and \(\Delta (p^{\prime }, q^{\prime }, r^{\prime })\) be its comparison triangle.

Let \(\alpha , \beta , \gamma \) (resp. \(\alpha ^{\prime }, \beta ^{\prime }, \gamma ^{\prime })\) be the angles of \(\Delta (p,q,r)\) (resp. \(\Delta (p^{\prime }, q^{\prime }, r^{\prime }))\) at the vertices p,q,r (resp. \(p^{\prime }, q^{\prime }, r^{\prime })\) . Then, the following inequalities hold:

Let z be a point in the geodesic joining p to q and \(z^{\prime }\) its comparison point in the interval \([p^{\prime }, q^{\prime }].\) Suppose that \(d(z,p)=\Vert z^{\prime }-p^{\prime }\Vert \) and \(d(z^{\prime },q^{\prime })=\Vert z^{\prime }-q^{\prime }\Vert \) . Then, the following inequality holds:

[ 25 ] Let \(x_0 \in \mathbb {P}\) and \(\{x_n\} \subset \mathbb {P}\) be such that \(x_n \rightarrow x_0.\) Then, for any \(y \in \mathbb {P},\) we have \(\exp _{x_n}^{-1}y \rightarrow \exp _{x_0}^{-1}y\) and \(\exp _{y}^{-1}x_n \rightarrow \exp _{y}^{-1}x_0.\)

The following propositions (see [ 17 ]) are very useful in our convergence analysis:

Proposition 3

Let \(\mathbb {P}\) be a Hadamard manifold and \(d: \mathbb {P} \times \mathbb {P}: \rightarrow \mathbb {R}\) be the distance function. Then the function d is convex with respect to the product Riemannian metric. In other words, given any pair of geodesics \(\gamma _1: [0,1] \rightarrow \mathbb {P}\) and \(\gamma _2: [0,1] \rightarrow \mathbb {P},\) then for all \(t \in [0,1],\) we have

In particular, for each \(y \in \mathbb {P},\) the function \(d(\cdot ,y): \mathbb {P} \rightarrow \mathbb {R}\) is a convex function.

Proposition 4

Let \(\mathbb {P}\) be a Hadamard manifold and \(x \in \mathbb {P}\) . The map \(\Phi _x=d^2(x,y)\) satisfying the following:

\(\Phi _x\) is convex. Indeed, for any geodesic \(\gamma : [0,1] \rightarrow \mathbb {P}\) , the following inequality holds for all \(t \in [0,1]:\)

\(\Phi _x\) is smooth. Moreover, \(\partial \Phi _x(y)=-2\exp _{y}^{-1}x.\)

[ 18 ] Let \(\{v_n\}\) and \(\{\delta _n\}\) be nonnegative sequences which satisfy

Moreover, if \(\sum \limits _{n=1}^{\infty }\delta _n < + \infty ,\) then \(\{v_n\}\) is bounded.

[ 34 ] Let \(\{a_n\}, \{\varphi _n\}\) and \(\{\beta _n\}\) be nonnegative sequences which satisfy

If \(\sum \limits _{n=1}^{\infty }\beta _n< + \infty \) and \(\sum \limits _{n=1}^{\infty }\varphi _n < + \infty ,\) then \(\lim \limits _{n \rightarrow \infty }a_n\) exists.

3 Main Result

In this section, we present an iterative method for solving variational inclusion problem in the settings of Hadamard manifolds. We state the following assumptions:

Assumption 1

\(\Phi \in \mathcal {H(K)}\) is monotone and L -Lipschitz continuous, and \(\Psi \in \mathcal {X(K)}\) is maximal monotone.

The solution set \(\Omega := (\Phi +\Psi )^{-1}(\textbf{0})\) is nonempty.

\(\{\lambda _k\}\) is a nonnegative real numbers sequence such that \(\sum \limits _{k=1}^{\infty }\lambda _k< \infty .\)

We start by establishing a technical lemma useful to our analysis.

[ 2 , 27 ] Let \(\{q_k\}\) be a sequence generated by Algorithm 19 and the sequence \(\{\rho _{k}\}\) is generated by (19). Then we have that \(\lim \limits _{k \rightarrow \infty }\rho _{k}=\rho \) and \(\rho \in \bigg [\min \big \{\frac{\mu }{L},~ \rho _{0}\big \}, \rho _{0}+ \lambda \bigg ],\) where \(\lambda =\sum \limits _{k=0}^{\infty }\lambda _k.\)

It is obvious that the stepsize in Algorithm 19 is allowed to increase from iteration to iteration and so (19) reduces the dependence on the initial stepsize \(\rho _{0}\) . Also, since \(\{\lambda _k\}\) is summable, we obtain \(\lim \limits _{k \rightarrow \infty }\lambda _k=0.\) Thus the stepsize \(\lambda _{k}\) may be non-increasing when k is large. If \(\lambda _k \equiv 0,\) the step size in (19) reduces to the one in [ 20 ].

Suppose that Assumptions (L1)-(L3) holds and let \(\{q_k\}\) be a sequence generated by Algorithm 19. If \(\sum \nolimits _{k=1}^{\infty }\alpha _k < +\infty \) and \(\sum \nolimits _{k=1}^{\infty }\theta _k < + \infty ,\) then

\(d(q_{k+1}, p) \leqslant M \cdot \prod \limits _{j=1}^{k}(1+ 2(\alpha _j + \theta _j(1+\alpha _j))),\) where \(M:=\max \{d(q_1, p), d(q_2, p)\}.\)

The sequence \(\{q_k\}\) converges to an element in \(\Omega .\)

Let \(p \in \Omega ,\) then \(-\Phi (q) \in \Psi (p)\) . Using (16) of Algorithm 19, we get \(\frac{1}{\rho _k}\exp _{t_k}^{-1}z_k-\Gamma _{t_k,z_k}\Phi (z_k) \in \Psi (t_k).\) By applying the monotonicity of \(\Psi \) , we deduce that

Since \(\Phi \) is a monotone vector field, then

By combining ( 20 ) and ( 21 ), we have

Now, for \(k \in \mathbb {N}.\) Let \(\Delta (z_k, t_k, p) \subseteq \mathbb {P}\) be a geodesic triangle with vertices \(z_k, t_k\) and p and let \(\Delta (z_k^{\prime }, t_k^{\prime }, p^{\prime })\subset \mathbb {R}^2\) be the corresponding comparison triangle, thus we have from Lemma 5 (ii) that \(d(z_k, p)=\Vert z_k^{\prime }-p^{\prime }\Vert ,~ d(t_k, p)=\Vert t_k^{\prime }-p^{\prime }\Vert \) and \(d(t_k^{\prime }, z_k^{\prime })=\Vert t_k^{\prime }-z_k^{\prime }\Vert \) . Also, let \(\Delta (q_{k+1}, t_k, p) \subseteq \mathbb {P}\) be a geodesic triangle with vertices \(q_{k+1},~ t_k\) and p , then \(\Delta (q_{k+1}^{\prime }, t_k^{\prime }, p^{\prime }) \subseteq \mathbb {R}^2\) is the corresponding comparison triangle. Hence, we have \(d(q_{k+1}, p)=\Vert q_{k+1}^{\prime }-p^{\prime }\Vert , ~d(t_k, p)=\Vert t_k^{\prime }-p^{\prime }\Vert \) and \(d(q_{k+1}, t_k)=\Vert q_{k+1}^{\prime }-t_k^{\prime }\Vert .\)

Let r and \(r^{\prime }\) be the angles of the vertices \(t_k\) and \(t_k^{\prime }\) , respectively. By Lemma 5 (i), we get \(r^{\prime } \geqslant r\) . Therefore, we obtain from Lemma 4 and ( 13 ) that

Following the same argument as in ( 24 ), we have

Hence, we deduce from (18) that

On substituting ( 24 ), ( 25 ) and ( 26 ) into ( 23 ), we obtain

Using Remark 1 , Lemma 2 and ( 27 ), we get

which also implies that

It follows from the definition of \(q_{k+1}\) that \(\exp _{t_k}^{-1}q_{k+1}=\rho _k(\Gamma _{t_k,z_k}\Phi (z_k)-\Phi (t_k)).\) Using the last inequality, we obtain that

By substituting (19) and ( 22 ) in ( 30 ), we get

By utilizing the geodesic triangles \(\bigtriangleup (w_k, q_k, p) \subset \mathbb {P}\) and \(\bigtriangleup (q_k, q_{k-1}, p) \subset \mathbb {P}\) with their respective comparison triangles \(\bigtriangleup (w_k^{\prime }, q_{k}^{\prime }, p^{\prime }) \subseteq \mathbb {R}^2\) . Then, by Lemma 5 (ii), we have \(d(w_k, q_k)= \Vert w_k^{\prime }-q_k^{\prime }\Vert ,~ d(w_k, p)=\Vert w_k^{\prime }-p^{\prime }\Vert \) and \(d(q_{k}, q_{k-1})=\Vert q_{k}^{\prime }-q_{k-1}^{\prime }\Vert \) . Similarly, using the geodesic triangles \(\bigtriangleup (z_k, w_k, p) \subset \mathbb {P}\) and \(\bigtriangleup (q_k, q_{k-1}, p) \subset \mathbb {P}\) with their respective comparison triangle \(\bigtriangleup (z_{k}^{\prime }, w_{k}^{\prime }, p^{\prime }) \subseteq \mathbb {R}^2\) . Then, by Lemma 5 (ii), we have \(d(z_k, w_k)=\Vert z_k^{\prime }-w_{k}^{\prime }\Vert ,~ d(z_k, q_k)=\Vert z_{k}^{\prime }-q_{k}^{\prime }\Vert \) and \(d(z_k, p)=\Vert z_{k}^{\prime }-p^{\prime }\Vert \) . From step 1 of Algorithm 19, we have that \(w_k^{\prime }=q_k^{\prime } + \alpha _k(q_{k}^{\prime }-q_{k-1}^{\prime })\) and \(z_{k}^{\prime }=w_{k}^{\prime } + \theta _k(w_{k}^{\prime }-q_{k-1}^{\prime })\) , thus

Similarly, it is easy to see that

By definition of \(z_{k},\) ( 32 ) and ( 33 ), we get

Since \(\lim \limits _{k \rightarrow \infty }\big (1-\mu ^2\frac{\rho _k^{2}}{\rho _{k+1}^2}\big ) =1-\mu ^2 > 0,\) this implies that there exists \(N> 0\) such that \(1-\mu ^2 \frac{\rho _k^{2}}{\rho _{k+1}^2} > 0, \forall ~k \geqslant \mathbb {N}.\)

From ( 31 ) and ( 34 ), we deduce that

By applying Lemma 7 , we obtain that

where \(M=\max \{d(q_1, p), d(q_2, p)\}\) . Hence, the proof completes.

To establish the second part of the proof, we need to show that \(\{q_k\}\) converges to a point in \(\Omega .\) Since \(\sum \limits _{k=1}^{\infty }\alpha _k < + \infty \) and \(\sum \limits _{k=1}^{\infty }\theta _k < + \infty \) , by Lemma 7 and ( 36 ), the sequence \(\{q_k\}\) is bounded. This also implies that \(\sum \limits _{k=1}^{\infty }\alpha _k d(q_{k},q_{k-1}) < + \infty \) and \(\sum \limits _{k=1}^{\infty }\theta _k d(q_{k},q_{k-1}) < + \infty .\) Using Lemma 8 in ( 35 ), we can claim that \(\lim \limits _{k \rightarrow \infty } d(q_{k}, p)\) exists. We have from Lemma 5 (ii) and Proposition 4 that

We also consider

But from ( 14 ), we have

On substituting ( 39 ) into ( 38 ), we get

We deduce from Lemma 5 , ( 37 ) and ( 40 ) that

On substituting ( 41 ) into ( 31 ), we obtain

The last inequality yields

Since \(\lim \limits _{k \rightarrow \infty }d(q_{k}, p)\) exists, \(\sum \limits _{k=1}^{\infty }\alpha _k< + \infty \) and \(\sum \limits _{k=1}^{\infty }\theta _k < + \infty \) . It follows from ( 43 ) that

From ( 45 ), we get

From ( 44 ) and ( 46 ), we have

Using ( 45 ) and ( 46 ), we deduce that

Since \(\{q_k\}\) is bounded, there exists a subsequence \(\{q_{k_l}\}\) which converges to a cluster point \(\overline{p}\) . Also, from ( 47 ), there exists a subsequence \(\{t_{k_l}\}\) of \(\{t_k\}\) which converges weakly to \(\overline{p} \in \mathbb {P}\) . By (17), we deduce that

Thus, by applying ( 44 ), we have

Since \(\Phi \) is a Lipschitz continuous vector field and \(z_{k_{l}} \rightarrow \overline{p}\) as \(l \rightarrow \infty .\) Combining ( 49 ) and ( 50 ), we obtain

Also, using the fact that \(\Psi \) is a maximal monotone vector field, so it is upper Kuratowski semicontinuous. Thus \(-\Gamma (\overline{p}) \in \Psi (\overline{p}),\) which implies that \(\overline{p}\) solves \(\Omega \) . Lastly by Lemma 1 , we obtain that \(\{q_k\}\) converges to a point in \(\Omega .\)

4 Numerical Example

Let \(\mathbb {R}_{++}=\{x \in \mathbb {R}: x > 0 \}\) and \(\mathbb {P}=(\mathbb {R}_{++}, \langle \cdot ,\cdot \rangle )\) be the Riemannian manifold with Riemannian metric defined by \(\langle p,q\rangle =\frac{1}{x^2}pq\; \in \mathbb {R}_{++}, \; p,q \in T_{x}\mathbb {P}.\) The Riemannian distance \(d: \mathbb {P} \times \mathbb {P} \rightarrow \mathbb {R}_{+}\) is given by \(d(x,y)=|\ln \frac{y}{x}|\) for all \(x,y \in \mathbb {P}.\) Let \(x \in \mathbb {P},\) then the exponential map \(\exp _{x}:T_x \mathbb {P} \rightarrow \mathbb {P}\) is defined by \(\exp _{x} sq=x\textrm{e}^{\frac{qs}{x}}\) for all \(q \in T_x \mathbb {P}.\) The inverse of the exponential map, \(\exp _{x}^{-1}: \mathbb {P} \rightarrow T_x \mathbb {P}\) is defined by \(\exp _{x}^{-1}y=x\ln \frac{y}{x}\) for all \(x,y \in \mathbb {P}.\) The parallel transport is the identity on \(T\mathbb {P}.\) Let \(\mathcal {K}=(0,1],\) \(\Psi :\mathcal {K} \rightarrow \mathbb {R}\) and \(\Phi : \mathcal {K} \rightarrow T\mathbb {P}\) be defined by \(\Psi (x)=x\ln x\) and \(\Phi (x)=x(1+\ln x)\) , respectively. Then, \(\Psi \) is maximal monotone on \(\mathcal {K}\) and \(\Phi \) is a continuous and monotone vector field on \(\mathcal {K}.\) By simple calculation, we obtain that \(t_k\) in Algorithm 19 can be expressed as

and \((\Phi +\Psi )^{-1}(0)=\frac{1}{\sqrt{\textrm{e}}}.\) We choose \(\alpha _{k}=\frac{1}{k+1},\) \(\theta _k=\frac{1}{2n+3},\) \(\lambda _k=\frac{1}{k\sqrt{k}},\) \(\mu =\frac{1}{2}\) and \(\rho _{0}=0.3.\) We terminate the execution of the process at \(E_k=d(x_{k+1},x_k)=10^{-3}\) and make a comparison of Algorithm 19 with a step inertial and non-accelerated versions of the Algorithm. We test the convergence of the method with some initial values of \(x_0\) and \(x_1.\) The result of this experiment is shown in Fig.  1 .

\(x_0=0.1\) and \(x_1=0.18.\)

\(x_0=0.9\) and \(x_1=0.5.\)

Numerical report for Example 2

Let \(\mathbb {R}^{3}_{++}=\{x=(x_1,x_2,x_3) \in \mathbb {R}^3: x_i >0, i=1,2,3\},\) \(\mathbb {P}=(\mathbb {R}^{3}_{++},\langle \cdot ,\cdot \rangle ) \) be the Riemannian manifold with the Riemannian metric is defined by

where G ( x ) is a diagonal matrix defined \(G(x)=\textrm{diag}(x_{1}^{-2},x_{2}^{-2},x_{3}^{-2}).\) The Riemannian \(d: \mathbb {P} \times \mathbb {P} \rightarrow \mathbb {R}_{+}\) is defined by

The sectional curvature of the Riemannian manifold \(\mathbb {P}\) is 0. Thus \(\mathbb {P}=(\mathbb {R}_{++}^{3},\langle \cdot ,\cdot \rangle )\) is a Hadamard manifold. Let \(x=(x_1,x_2,x_3) \in \mathbb {P}.\) Then, the exponential map \(\exp _{x}: T_x \mathbb {P} \rightarrow \mathbb {P}\) is defined by

for all \(p=(p_1,p_p,p_3) \in T_x \mathbb {P}.\) The inverse of the exponential map, \(\exp _{x}^{-1}: \mathbb {P}\rightarrow T_x \mathbb {P}\) is defined by

for all \(x,y \in \mathbb {P}.\) The parallel transport \(\Gamma _{y,x}: T_x \mathbb {P} \rightarrow T_y \mathbb {P}\) is defined by

for all \(p=(p_1,p_2,p_3) \in T_x \mathbb {P}.\) Let \(\mathcal {K}=\{x=(x_1,x_2,x_3) \in \mathbb {P}: 0 < x_i \leqslant 1, ~~\text{ for }~~i=1,2,3 \}\) be the geodesic convex subset of \(\mathbb {P}.\) Let \(\Phi :\mathbb {M} \rightarrow T\mathbb {P}\) be defined by

and \(\Phi :\mathbb {M} \rightarrow T\mathbb {P}\) be defined by

Then, \(\Psi \) is maximal monotone vector field on \(\mathcal {K}\) and \(\Phi \) is continuous and monotone vector field on \(\mathcal {K}\) (see [ 8 , Example 1]). By simple calculation, we see that \(t_k\) in Algorithm 19 can be expressed as

Note that \((\Psi +\Phi )^{-1}(0)=\{(1,\frac{1}{\textrm{e}},\frac{1}{2})\}.\) Let \(\alpha _{k}=\frac{1}{k+1},\) \(\theta =\frac{1}{2k+3},\) \(\lambda _k=\frac{1}{k\sqrt{k}},\) \(\mu =\frac{1}{2}\) and \(\rho _0=0.9.\) We terminate the execution of the process at \(E_k=d(x_{k+1},x_k)=10^{-4}\) and make a comparison of Algorithm 19 with one inertial and a non-accelerated versions of the Algorithm. The result of this experiment is shown in Fig.  2 for two initial values of \(x_0\) and \(x_1.\)

\(x_0=[1.5,1.5,1.5]'\) and \(x_1=[1.3,1.2,1.1]'.\)

\(x_0=[1.8,1.8,1.8]'\) and \(x_1=[1.5,1.5,1.6]'.\)

5 Conclusion

In this manuscript, we proposed double inertial methods with a forward–backward method for solving variational inclusion problem in the settings of a Hadamard manifold. We establish a convergence result for solving variational inclusion problem and illustrate some numerical examples to show the performance of our method in comparison with some related ones in the literature. It can be seen from our figures that the two steps inertial extrapolation method illustrated in our manuscript converges faster that the one step inertial method and the non-inertial iterative method. This result discussed in this manuscript is new in the settings of a Hadamard manifold.

Data Availability

No data were used for the research described in the article.

Abass, H.A., Izuchukwu, C., Mewomo, O.T., Dong, Q.L.: Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach spaces. Fixed Point Theory 21 (2), 397–412 (2020)

Article   MathSciNet   Google Scholar  

Abass, H.A., Oyewole, O.K., Jolaoso, L.O., Aremu, K.O.: Modified inertial Tseng method for solving variational inclusion and fixed point problems on Hadamard manifolds. Appl. Anal. (2023). https://doi.org/10.1080/00036811.2023.2256357

Article   Google Scholar  

Abass, H.A., Oyewole, O.K., Onifade, O.M., Narain, O.K.: Iterative approximation of common solution of variational inequality and certain optimization problems with multiple output sets in Hadamard space. Analysis (2024). https://doi.org/10.1515/anly-2022-1075

Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38 (4), 1102–1119 (2000)

Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9 , 3–11 (2001)

Antipin, A.S.: Minimization of convex functions on convex sets by means of differential equations. Diff. Equ. 30 (9), 1365–1375 (1994)

MathSciNet   Google Scholar  

Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov’s accelerated forward-backward method os actually faster than \(\frac{1}{k^2}\) . SIAM J. Optim. 26 , 1824–1834 (2016)

Sahu, D.R., Babu, F., Sharma, S.: A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications. Numer Algor 94 , 1435–1460 (2023)

Baćak, M.: Old and new challenges in Hadamard spaces. arXiv:1807.01355v2 (2018)

Baćak, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24 , 1542–1566 (2014)

Baćak, M.: The proximal point algorithm in metric spaces. Israel. J. Math. 194 , 689–701 (2013)

Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 319. Springer, Berlin (1999)

Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 , 1168–1200 (2005)

Cruz, J.X., Ferreira, O.P., Pérez, L.R., et al.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5 (1), 69–79 (2000)

Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraints. Commun. Pure Appl. Math. 57 , 1413–1457 (2004)

Dong, Q.L., Cho, Y.J., Rassias, T.M.: General inertial Mann algorithms and their convergence analysis for nonexpansive mappings. Appl. Nonlinear Anal., pp. 175-191 (2018)

Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51 (2), 257–270 (2002)

Hanjing, A., Suantai, S.: A fast image restoration algorithm based on a fixed point and optimization method. Mathematics 8 (3), 378 (2020)

Jabeen, S., Noor, M.A., Noor, K.I.: Inertial methods for solving system of quasi variational inequalities. J. Adv. Math. Stud. 15 (1), 1–10 (2022)

Khammahawong, K., Kumam, P., Chaipunya, P., Martinez, J.M.: Tseng’s method for inclusion problems on Hadamard manifolds. Optimization 71 (15), 4367–4401 (2022)

Khammahawong, K., Chaipunya, P., Kumam, P.: An inertial Mann algorithm for nonexpansive for nonexpansive mappings on Hadamard manifolds. AIMS Math. 8 (1), 2093–2116 (2023)

Li, C., López, G., Màrquez, V.M.: Resolvent of set-valued monotone vector fields in Hadamard manifolds. J. Set-Valued Anal. 19 , 361–383 (2011)

Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwan. J. Math 14 , 541–559 (2010)

Li, X., Dong, Q.L., Gibali, A.: PMICA, Parallel multi-step inertial contracting algorithm for solving common variational inclusion. J. Nonlinear Funct. Anal. 2022 , 7 (2022)

Google Scholar  

Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on hadamard manifolds. J. Lond. Math. Soc. 79 (3), 663–683 (2009)

Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 , 964–979 (1979)

Liu, H., Yang, J.: Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comp. Optim. Appl. 77 , 491–508 (2020)

López, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal., Article ID 109236 (2102)

Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51 , 311–325 (2015)

Liang, J.: Convergence rates of first-order operator splitting methods. PhD thesis, Normandie Universite, GREYC CNRS UMR 6072 (2016)

Mebawondu, A.A., Abass, H.A., Oyewole, O.K., Narain, O.K.: Generalized split null point of sum of monotone operators in Hilbert spaces. Demonstr. Math. 54 (1), 359–376 (2021)

Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155 , 447–454 (2003)

Nestrov, Y.: A method for solving the convex programming problem with convergence rate \(O(\frac{1}{k^2})\) . Dokl. Akad. Nauk SSSR 269 , 543–547 (1983)

Osilike, M.O., Aniagbosor, S.C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 32 (10), 1181–1191 (2000)

Oyewole, O.K., Reich, S.: An inertial subgradient extragradient method for approximating solution to equilibrium problems in Hadamard manifold. Axioms 12 (3), 256 (2023)

Oyewole, O.K., Abass, H.A., Mebawondu, A.A., Aremu, K.O.: A Tseng extragradient method for solving variational variational inequality problems in Banach spaces. Numer. Algorithms 89 (2), 769–789 (2022)

Polyak, B.T.: Introduction to Optimization. Optimization Software, Publications Division, New York (1987)

Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh Vychisl. Mat. Mat. Fiz. 4 , 1–17 (1964)

Poon, C., Liang, J.: Geometry of first-order methods and adaptive acceleration. arxiv: 2003.03910 (2020)

Sakai, T.: Riemannian geometry. Translations of mathematical monographs, Vol. 149, Providence (RI): American Mathematical Society (1996)

Shehu, Y.: Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces. Results Math. 74 , 138 (2018)

Shehu, Y., Yao, J.C.: Rate of convergence for inertial iterative method for countable family of certain quasi-nonexpansive mappings. J. Nonlinear Convex Anal. 21 , 533–541 (2020)

Shehu, Y., Ezeora, J.N.: Weak and linear convergence of a generalized proximal point algorithm with alternating inertial steps for a monotone inclusion problem. J. Nonlinear Var. Anal. 5 , 881–892 (2021)

Sra, S., Nowozin, S., Wright, S.J.: Optimization for Machine Learning. MIT Press, Cambridge (2012)

Suantai, S., Inkrong, P., Cholamjiak, P.: Forward-backward-forward algorithms involving two inertial terms for monotone inclusions. Comput. Appl. Math. 42 , 255 (2023)

Thong, D.V., Cholamjiak, P.: Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions. Comput. Appl. Math. 38 , 94 (2019)

Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58 , 267–288 (1996)

Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38 , 431–446 (2009)

Ugwunnadi, G.C., Abass, H.A., Aphane, M., Oyewole, O.K.: Inertial Halpern-typemethod for solving split feasibility and fixed point problems via dynamical stepsize in real Banach spaces. Annali dell‘Universita di Ferrara (2023). https://doi.org/10.1007/s11565-023-00473-6

Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim Theory Appl. 146 , 691–708 (2010)

Yao, Y., Iyiola, O.S., Shehu, Y.: Subgradient extragradient method with double inertial steps for variational inequalities. J. Sci. Comput. 90 (2), 71 (2022)

Zhao, X., Yao, J.C., Yao, Y.: A proximal algorithm for solving split monotone variational inclusions. U.P.B. Sci. Bull. Ser. A 82 (3), 43–52 (2020)

Download references

Open access funding provided by Sefako Makgatho Health Sciences University.

Author information

Authors and affiliations.

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Science University, Pretoria, 0204, South Africa

Hammed Anuoluwapo Abass, Kazeem Olalekan Aremu & Lateef Olakunle Jolaoso

Department of Mathematics, The Technion-Israel Institute of Technology, Haifa, 32000, Israel

Olawale Kazeem Oyewole

Department of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto, Nigeria

Kazeem Olalekan Aremu

Department of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom

Lateef Olakunle Jolaoso

You can also search for this author in PubMed   Google Scholar

Contributions

H. A. Abass contributed to the conceptualization; H. A. Abass and O. K. Oyewole were involved in the methodology; O. K. Oyewole contributed to the software; H. A. Abass, L. O. Jolaoso and K. O. Aremu assisted in the validation; K. O. Aremu and L. O. Jolaoso performed the formal analysis; H. A. Abass contributed to writing–original draft preparation; all authors contributed to writing–review and editing and supervision; H. A. Abass contributed to the project administration. All authors have contributed equally to the completion of this manuscript.

Corresponding author

Correspondence to Hammed Anuoluwapo Abass .

Ethics declarations

Competing interests.

The authors declare that there is no competing interest.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

About this article

Abass, H.A., Oyewole, O.K., Aremu, K.O. et al. Self-adaptive Technique with Double Inertial Steps for Inclusion Problem on Hadamard Manifolds. J. Oper. Res. Soc. China (2024). https://doi.org/10.1007/s40305-024-00537-0

Download citation

Received : 08 September 2023

Revised : 04 January 2024

Accepted : 07 January 2024

Published : 02 April 2024

DOI : https://doi.org/10.1007/s40305-024-00537-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Variational inclusion problem
• Double inertial method
• Hadamard manifold
• Monotone operator
• Riemannian manifold

Mathematics Subject Classification

• Find a journal
• Publish with us
• Track your research