sample of problem solving

26 Expert-Backed Problem Solving Examples – Interview Answers

Published: February 13, 2023

Interview Questions and Answers

Actionable advice from real experts:

Biron Clark

Former Recruiter

Contributor

Dr. Kyle Elliott

Career Coach

Hayley Jukes

Editor-in-Chief

Biron Clark , Former Recruiter

Kyle Elliott , Career Coach

Hayley Jukes , Editor

As a recruiter , I know employers like to hire people who can solve problems and work well under pressure.

 A job rarely goes 100% according to plan, so hiring managers are more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical.

But how do they measure this?

Hiring managers will ask you interview questions about your problem-solving skills, and they might also look for examples of problem-solving on your resume and cover letter. 

In this article, I’m going to share a list of problem-solving examples and sample interview answers to questions like, “Give an example of a time you used logic to solve a problem?” and “Describe a time when you had to solve a problem without managerial input. How did you handle it, and what was the result?”

• Problem-solving involves identifying, prioritizing, analyzing, and solving problems using a variety of skills like critical thinking, creativity, decision making, and communication.
• Describe the Situation, Task, Action, and Result ( STAR method ) when discussing your problem-solving experiences.
• Tailor your interview answer with the specific skills and qualifications outlined in the job description.
• Provide numerical data or metrics to demonstrate the tangible impact of your problem-solving efforts.

What are Problem Solving Skills? 

Problem-solving is the ability to identify a problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation. 

Problem-solving encompasses other skills that can be showcased in an interview response and your resume. Problem-solving skills examples include:

• Critical thinking
• Analytical skills
• Decision making
• Research skills
• Technical skills
• Communication skills
• Adaptability and flexibility

Why is Problem Solving Important in the Workplace?

Problem-solving is essential in the workplace because it directly impacts productivity and efficiency. Whenever you encounter a problem, tackling it head-on prevents minor issues from escalating into bigger ones that could disrupt the entire workflow. 

Beyond maintaining smooth operations, your ability to solve problems fosters innovation. It encourages you to think creatively, finding better ways to achieve goals, which keeps the business competitive and pushes the boundaries of what you can achieve. 

Effective problem-solving also contributes to a healthier work environment; it reduces stress by providing clear strategies for overcoming obstacles and builds confidence within teams. 

Examples of Problem-Solving in the Workplace

• Correcting a mistake at work, whether it was made by you or someone else
• Overcoming a delay at work through problem solving and communication
• Resolving an issue with a difficult or upset customer
• Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
• Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
• Troubleshooting and resolving technical issues
• Handling and resolving a conflict with a coworker
• Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
• Taking initiative when another team member overlooked or missed something important
• Taking initiative to meet with your superior to discuss a problem before it became potentially worse
• Solving a safety issue at work or reporting the issue to those who could solve it
• Using problem solving abilities to reduce/eliminate a company expense
• Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
• Changing how a process, team, or task is organized to make it more efficient
• Using creative thinking to come up with a solution that the company hasn’t used before
• Performing research to collect data and information to find a new solution to a problem
• Boosting a company or team’s performance by improving some aspect of communication among employees
• Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

Problem-Solving Examples for Recent Grads/Entry-Level Job Seekers

• Coordinating work between team members in a class project
• Reassigning a missing team member’s work to other group members in a class project
• Adjusting your workflow on a project to accommodate a tight deadline
• Speaking to your professor to get help when you were struggling or unsure about a project
• Asking classmates, peers, or professors for help in an area of struggle
• Talking to your academic advisor to brainstorm solutions to a problem you were facing
• Researching solutions to an academic problem online, via Google or other methods
• Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

How To Answer “Tell Us About a Problem You Solved”

When you answer interview questions about problem-solving scenarios, or if you decide to demonstrate your problem-solving skills in a cover letter (which is a good idea any time the job description mentions problem-solving as a necessary skill), I recommend using the STAR method.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. 

Start by briefly describing the general situation and the task at hand. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact. Finally, describe the positive result you achieved.

Note: Our sample answers below are structured following the STAR formula. Be sure to check them out!

EXPERT ADVICE

Dr. Kyle Elliott , MPA, CHES Tech & Interview Career Coach caffeinatedkyle.com

How can I communicate complex problem-solving experiences clearly and succinctly?

Before answering any interview question, it’s important to understand why the interviewer is asking the question in the first place.

When it comes to questions about your complex problem-solving experiences, for example, the interviewer likely wants to know about your leadership acumen, collaboration abilities, and communication skills, not the problem itself.

Therefore, your answer should be focused on highlighting how you excelled in each of these areas, not diving into the weeds of the problem itself, which is a common mistake less-experienced interviewees often make.

Tailoring Your Answer Based on the Skills Mentioned in the Job Description

As a recruiter, one of the top tips I can give you when responding to the prompt “Tell us about a problem you solved,” is to tailor your answer to the specific skills and qualifications outlined in the job description. 

Once you’ve pinpointed the skills and key competencies the employer is seeking, craft your response to highlight experiences where you successfully utilized or developed those particular abilities. 

For instance, if the job requires strong leadership skills, focus on a problem-solving scenario where you took charge and effectively guided a team toward resolution. 

By aligning your answer with the desired skills outlined in the job description, you demonstrate your suitability for the role and show the employer that you understand their needs.

Amanda Augustine expands on this by saying:

“Showcase the specific skills you used to solve the problem. Did it require critical thinking, analytical abilities, or strong collaboration? Highlight the relevant skills the employer is seeking.”  

Interview Answers to “Tell Me About a Time You Solved a Problem”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” or “Tell me about a time you solved a problem,” since you’re likely to hear different versions of this interview question in all sorts of industries.

The example interview responses are structured using the STAR method and are categorized into the top 5 key problem-solving skills recruiters look for in a candidate.

1. Analytical Thinking

Situation: In my previous role as a data analyst , our team encountered a significant drop in website traffic.

Task: I was tasked with identifying the root cause of the decrease.

Action: I conducted a thorough analysis of website metrics, including traffic sources, user demographics, and page performance. Through my analysis, I discovered a technical issue with our website’s loading speed, causing users to bounce. 

Result: By optimizing server response time, compressing images, and minimizing redirects, we saw a 20% increase in traffic within two weeks.

2. Critical Thinking

Situation: During a project deadline crunch, our team encountered a major technical issue that threatened to derail our progress.

Task: My task was to assess the situation and devise a solution quickly.

Action: I immediately convened a meeting with the team to brainstorm potential solutions. Instead of panicking, I encouraged everyone to think outside the box and consider unconventional approaches. We analyzed the problem from different angles and weighed the pros and cons of each solution.

Result: By devising a workaround solution, we were able to meet the project deadline, avoiding potential delays that could have cost the company $100,000 in penalties for missing contractual obligations.

3. Decision Making

Situation: As a project manager , I was faced with a dilemma when two key team members had conflicting opinions on the project direction.

Task: My task was to make a decisive choice that would align with the project goals and maintain team cohesion.

Action: I scheduled a meeting with both team members to understand their perspectives in detail. I listened actively, asked probing questions, and encouraged open dialogue. After carefully weighing the pros and cons of each approach, I made a decision that incorporated elements from both viewpoints.

Result: The decision I made not only resolved the immediate conflict but also led to a stronger sense of collaboration within the team. By valuing input from all team members and making a well-informed decision, we were able to achieve our project objectives efficiently.

4. Communication (Teamwork)

Situation: During a cross-functional project, miscommunication between departments was causing delays and misunderstandings.

Task: My task was to improve communication channels and foster better teamwork among team members.

Action: I initiated regular cross-departmental meetings to ensure that everyone was on the same page regarding project goals and timelines. I also implemented a centralized communication platform where team members could share updates, ask questions, and collaborate more effectively.

Result: Streamlining workflows and improving communication channels led to a 30% reduction in project completion time, saving the company $25,000 in operational costs.

5. Persistence 

Situation: During a challenging sales quarter, I encountered numerous rejections and setbacks while trying to close a major client deal.

Task: My task was to persistently pursue the client and overcome obstacles to secure the deal.

Action: I maintained regular communication with the client, addressing their concerns and demonstrating the value proposition of our product. Despite facing multiple rejections, I remained persistent and resilient, adjusting my approach based on feedback and market dynamics.

Result: After months of perseverance, I successfully closed the deal with the client. By closing the major client deal, I exceeded quarterly sales targets by 25%, resulting in a revenue increase of $250,000 for the company.

Tips to Improve Your Problem-Solving Skills

Throughout your career, being able to showcase and effectively communicate your problem-solving skills gives you more leverage in achieving better jobs and earning more money .

So to improve your problem-solving skills, I recommend always analyzing a problem and situation before acting.

 When discussing problem-solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Don’t just say you’re good at solving problems. Show it with specifics. How much did you boost efficiency? Did you save the company money? Adding numbers can really make your achievements stand out.

To get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t.

Think about how you can improve researching and analyzing a situation, how you can get better at communicating, and deciding on the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem-solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem-solving ability.

More Interview Resources

• 3 Answers to “How Do You Handle Stress?”
• How to Answer “How Do You Handle Conflict?” (Interview Question)
• Sample Answers to “Tell Me About a Time You Failed”

About the Author

Biron Clark is a former executive recruiter who has worked individually with hundreds of job seekers, reviewed thousands of resumes and LinkedIn profiles, and recruited for top venture-backed startups and Fortune 500 companies. He has been advising job seekers since 2012 to think differently in their job search and land high-paying, competitive positions. Follow on Twitter and LinkedIn .

Read more articles by Biron Clark

About the Contributor

Kyle Elliott , career coach and mental health advocate, transforms his side hustle into a notable practice, aiding Silicon Valley professionals in maximizing potential. Follow Kyle on LinkedIn .

About the Editor

Hayley Jukes is the Editor-in-Chief at CareerSidekick with five years of experience creating engaging articles, books, and transcripts for diverse platforms and audiences.

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39 Best Problem-Solving Examples

Problem-solving is a process where you’re tasked with identifying an issue and coming up with the most practical and effective solution.

This indispensable skill is necessary in several aspects of life, from personal relationships to education to business decisions.

Problem-solving aptitude boosts rational thinking, creativity, and the ability to cooperate with others. It’s also considered essential in 21st Century workplaces.

If explaining your problem-solving skills in an interview, remember that the employer is trying to determine your ability to handle difficulties. Focus on explaining exactly how you solve problems, including by introducing your thoughts on some of the following frameworks and how you’ve applied them in the past.

Problem-Solving Examples

1. divergent thinking.

Divergent thinking refers to the process of coming up with multiple different answers to a single problem. It’s the opposite of convergent thinking, which would involve coming up with a singular answer .

The benefit of a divergent thinking approach is that it can help us achieve blue skies thinking – it lets us generate several possible solutions that we can then critique and analyze .

In the realm of problem-solving, divergent thinking acts as the initial spark. You’re working to create an array of potential solutions, even those that seem outwardly unrelated or unconventional, to get your brain turning and unlock out-of-the-box ideas.

This process paves the way for the decision-making stage, where the most promising ideas are selected and refined.

Go Deeper: Divervent Thinking Examples

2. Convergent Thinking

Next comes convergent thinking, the process of narrowing down multiple possibilities to arrive at a single solution.

This involves using your analytical skills to identify the best, most practical, or most economical solution from the pool of ideas that you generated in the divergent thinking stage.

In a way, convergent thinking shapes the “roadmap” to solve a problem after divergent thinking has supplied the “destinations.”

Have a think about which of these problem-solving skills you’re more adept at: divergent or convergent thinking?

Go Deeper: Convergent Thinking Examples

3. Brainstorming

Brainstorming is a group activity designed to generate a multitude of ideas regarding a specific problem. It’s divergent thinking as a group , which helps unlock even more possibilities.

A typical brainstorming session involves uninhibited and spontaneous ideation, encouraging participants to voice any possible solutions, no matter how unconventional they might appear.

It’s important in a brainstorming session to suspend judgment and be as inclusive as possible, allowing all participants to get involved.

By widening the scope of potential solutions, brainstorming allows better problem definition, more creative solutions, and helps to avoid thinking “traps” that might limit your perspective.

Go Deeper: Brainstorming Examples

4. Thinking Outside the Box

The concept of “thinking outside the box” encourages a shift in perspective, urging you to approach problems from an entirely new angle.

Rather than sticking to traditional methods and processes, it involves breaking away from conventional norms to cultivate unique solutions.

In problem-solving, this mindset can bypass established hurdles and bring you to fresh ideas that might otherwise remain undiscovered.

Think of it as going off the beaten track when regular routes present roadblocks to effective resolution.

5. Case Study Analysis

Analyzing case studies involves a detailed examination of real-life situations that bear relevance to the current problem at hand.

For example, if you’re facing a problem, you could go to another environment that has faced a similar problem and examine how they solved it. You’d then bring the insights from that case study back to your own problem.

This approach provides a practical backdrop against which theories and assumptions can be tested, offering valuable insights into how similar problems have been approached and resolved in the past.

See a Broader Range of Analysis Examples Here

6. Action Research

Action research involves a repetitive process of identifying a problem, formulating a plan to address it, implementing the plan, and then analyzing the results. It’s common in educational research contexts.

The objective is to promote continuous learning and improvement through reflection and action. You conduct research into your problem, attempt to apply a solution, then assess how well the solution worked. This becomes an iterative process of continual improvement over time.

For problem-solving, this method offers a way to test solutions in real-time and allows for changes and refinements along the way, based on feedback or observed outcomes. It’s a form of active problem-solving that integrates lessons learned into the next cycle of action.

Go Deeper: Action Research Examples

7. Information Gathering

Fundamental to solving any problem is the process of information gathering.

This involves collecting relevant data , facts, and details about the issue at hand, significantly aiding in the understanding and conceptualization of the problem.

In problem-solving, information gathering underpins every decision you make.

This process ensures your actions are based on concrete information and evidence, allowing for an informed approach to tackle the problem effectively.

8. Seeking Advice

Seeking advice implies turning to knowledgeable and experienced individuals or entities to gain insights on problem-solving.

It could include mentors, industry experts, peers, or even specialized literature.

The value in this process lies in leveraging different perspectives and proven strategies when dealing with a problem. Moreover, it aids you in avoiding pitfalls, saving time, and learning from others’ experiences.

9. Creative Thinking

Creative thinking refers to the ability to perceive a problem in a new way, identify unconventional patterns, or produce original solutions.

It encourages innovation and uniqueness, often leading to the most effective results.

When applied to problem-solving, creative thinking can help you break free from traditional constraints, ideal for potentially complex or unusual problems.

Go Deeper: Creative Thinking Examples

10. Conflict Resolution

Conflict resolution is a strategy developed to resolve disagreements and arguments, often involving communication, negotiation, and compromise.

When employed as a problem-solving technique, it can diffuse tension, clear bottlenecks, and create a collaborative environment.

Effective conflict resolution ensures that differing views or disagreements do not become roadblocks in the process of problem-solving.

Go Deeper: Conflict Resolution Examples

11. Addressing Bottlenecks

Bottlenecks refer to obstacles or hindrances that slow down or even halt a process.

In problem-solving, addressing bottlenecks involves identifying these impediments and finding ways to eliminate them.

This effort not only smooths the path to resolution but also enhances the overall efficiency of the problem-solving process.

For example, if your workflow is not working well, you’d go to the bottleneck – that one point that is most time consuming – and focus on that. Once you ‘break’ this bottleneck, the entire process will run more smoothly.

12. Market Research

Market research involves gathering and analyzing information about target markets, consumers, and competitors.

In sales and marketing, this is one of the most effective problem-solving methods. The research collected from your market (e.g. from consumer surveys) generates data that can help identify market trends, customer preferences, and competitor strategies.

In this sense, it allows a company to make informed decisions, solve existing problems, and even predict and prevent future ones.

13. Root Cause Analysis

Root cause analysis is a method used to identify the origin or the fundamental reason for a problem.

Once the root cause is determined, you can implement corrective actions to prevent the problem from recurring.

As a problem-solving procedure, root cause analysis helps you to tackle the problem at its source, rather than dealing with its surface symptoms.

Go Deeper: Root Cause Analysis Examples

14. Mind Mapping

Mind mapping is a visual tool used to structure information, helping you better analyze, comprehend and generate new ideas.

By laying out your thoughts visually, it can lead you to solutions that might not have been apparent with linear thinking.

In problem-solving, mind mapping helps in organizing ideas and identifying connections between them, providing a holistic view of the situation and potential solutions.

15. Trial and Error

The trial and error method involves attempting various solutions until you find one that resolves the problem.

It’s an empirical technique that relies on practical actions instead of theories or rules.

In the context of problem-solving, trial and error allows you the flexibility to test different strategies in real situations, gaining insights about what works and what doesn’t.

16. SWOT Analysis

SWOT is an acronym standing for Strengths, Weaknesses, Opportunities, and Threats.

It’s an analytic framework used to evaluate these aspects in relation to a particular objective or problem.

In problem-solving, SWOT Analysis helps you to identify favorable and unfavorable internal and external factors. It helps to craft strategies that make best use of your strengths and opportunities, whilst addressing weaknesses and threats.

Go Deeper: SWOT Analysis Examples

17. Scenario Planning

Scenario planning is a strategic planning method used to make flexible long-term plans.

It involves imagining, and then planning for, multiple likely future scenarios.

By forecasting various directions a problem could take, scenario planning helps manage uncertainty and is an effective tool for problem-solving in volatile conditions.

18. Six Thinking Hats

The Six Thinking Hats is a concept devised by Edward de Bono that proposes six different directions or modes of thinking, symbolized by six different hat colors.

Each hat signifies a different perspective, encouraging you to switch ‘thinking modes’ as you switch hats. This method can help remove bias and broaden perspectives when dealing with a problem.

19. Decision Matrix Analysis

Decision Matrix Analysis is a technique that allows you to weigh different factors when faced with several possible solutions.

After listing down the options and determining the factors of importance, each option is scored based on each factor.

Revealing a clear winner that both serves your objectives and reflects your values, Decision Matrix Analysis grounds your problem-solving process in objectivity and comprehensiveness.

20. Pareto Analysis

Also known as the 80/20 rule, Pareto Analysis is a decision-making technique.

It’s based on the principle that 80% of problems are typically caused by 20% of the causes, making it a handy tool for identifying the most significant issues in a situation.

Using this analysis, you’re likely to direct your problem-solving efforts more effectively, tackling the root causes producing most of the problem’s impact.

21. Critical Thinking

Critical thinking refers to the ability to analyze facts to form a judgment objectively.

It involves logical, disciplined thinking that is clear, rational, open-minded, and informed by evidence.

For problem-solving, critical thinking helps evaluate options and decide the most effective solution. It ensures your decisions are grounded in reason and facts, and not biased or irrational assumptions.

Go Deeper: Critical Thinking Examples

22. Hypothesis Testing

Hypothesis testing usually involves formulating a claim, testing it against actual data, and deciding whether to accept or reject the claim based on the results.

In problem-solving, hypotheses often represent potential solutions. Hypothesis testing provides verification, giving a statistical basis for decision-making and problem resolution.

Usually, this will require research methods and a scientific approach to see whether the hypothesis stands up or not.

Go Deeper: Types of Hypothesis Testing

23. Cost-Benefit Analysis

A cost-benefit analysis (CBA) is a systematic process of weighing the pros and cons of different solutions in terms of their potential costs and benefits.

It allows you to measure the positive effects against the negatives and informs your problem-solving strategy.

By using CBA, you can identify which solution offers the greatest benefit for the least cost, significantly improving efficacy and efficiency in your problem-solving process.

Go Deeper: Cost-Benefit Analysis Examples

24. Simulation and Modeling

Simulations and models allow you to create a simplified replica of real-world systems to test outcomes under controlled conditions.

In problem-solving, you can broadly understand potential repercussions of different solutions before implementation.

It offers a cost-effective way to predict the impacts of your decisions, minimizing potential risks associated with various solutions.

25. Delphi Method

The Delphi Method is a structured communication technique used to gather expert opinions.

The method involves a group of experts who respond to questionnaires about a problem. The responses are aggregated and shared with the group, and the process repeats until a consensus is reached.

This method of problem solving can provide a diverse range of insights and solutions, shaped by the wisdom of a collective expert group.

26. Cross-functional Team Collaboration

Cross-functional team collaboration involves individuals from different departments or areas of expertise coming together to solve a common problem or achieve a shared goal.

When you bring diverse skills, knowledge, and perspectives to a problem, it can lead to a more comprehensive and innovative solution.

In problem-solving, this promotes communal thinking and ensures that solutions are inclusive and holistic, with various aspects of the problem being addressed.

27. Benchmarking

Benchmarking involves comparing one’s business processes and performance metrics to the best practices from other companies or industries.

In problem-solving, it allows you to identify gaps in your own processes, determine how others have solved similar problems, and apply those solutions that have proven to be successful.

It also allows you to compare yourself to the best (the benchmark) and assess where you’re not as good.

28. Pros-Cons Lists

A pro-con analysis aids in problem-solving by weighing the advantages (pros) and disadvantages (cons) of various possible solutions.

This simple but powerful tool helps in making a balanced, informed decision.

When confronted with a problem, a pro-con analysis can guide you through the decision-making process, ensuring all possible outcomes and implications are scrutinized before arriving at the optimal solution. Thus, it helps to make the problem-solving process both methodical and comprehensive.

29. 5 Whys Analysis

The 5 Whys Analysis involves repeatedly asking the question ‘why’ (around five times) to peel away the layers of an issue and discover the root cause of a problem.

As a problem-solving technique, it enables you to delve into details that you might otherwise overlook and offers a simple, yet powerful, approach to uncover the origin of a problem.

For example, if your task is to find out why a product isn’t selling your first answer might be: “because customers don’t want it”, then you ask why again – “they don’t want it because it doesn’t solve their problem”, then why again – “because the product is missing a certain feature” … and so on, until you get to the root “why”.

30. Gap Analysis

Gap analysis entails comparing current performance with potential or desired performance.

You’re identifying the ‘gaps’, or the differences, between where you are and where you want to be.

In terms of problem-solving, a Gap Analysis can help identify key areas for improvement and design a roadmap of how to get from the current state to the desired one.

31. Design Thinking

Design thinking is a problem-solving approach that involves empathy, experimentation, and iteration.

The process focuses on understanding user needs, challenging assumptions , and redefining problems from a user-centric perspective.

In problem-solving, design thinking uncovers innovative solutions that may not have been initially apparent and ensures the solution is tailored to the needs of those affected by the issue.

32. Analogical Thinking

Analogical thinking involves the transfer of information from a particular subject (the analogue or source) to another particular subject (the target).

In problem-solving, you’re drawing parallels between similar situations and applying the problem-solving techniques used in one situation to the other.

Thus, it allows you to apply proven strategies to new, but related problems.

33. Lateral Thinking

Lateral thinking requires looking at a situation or problem from a unique, sometimes abstract, often non-sequential viewpoint.

Unlike traditional logical thinking methods, lateral thinking encourages you to employ creative and out-of-the-box techniques.

In solving problems, this type of thinking boosts ingenuity and drives innovation, often leading to novel and effective solutions.

Go Deeper: Lateral Thinking Examples

34. Flowcharting

Flowcharting is the process of visually mapping a process or procedure.

This form of diagram can show every step of a system, process, or workflow, enabling an easy tracking of the progress.

As a problem-solving tool, flowcharts help identify bottlenecks or inefficiencies in a process, guiding improved strategies and providing clarity on task ownership and process outcomes.

35. Multivoting

Multivoting, or N/3 voting, is a method where participants reduce a large list of ideas to a prioritized shortlist by casting multiple votes.

This voting system elevates the most preferred options for further consideration and decision-making.

As a problem-solving technique, multivoting allows a group to narrow options and focus on the most promising solutions, ensuring more effective and democratic decision-making.

36. Force Field Analysis

Force Field Analysis is a decision-making technique that identifies the forces for and against change when contemplating a decision.

The ‘forces’ represent the differing factors that can drive or hinder change.

In problem-solving, Force Field Analysis allows you to understand the entirety of the context, favoring a balanced view over a one-sided perspective. A comprehensive view of all the forces at play can lead to better-informed problem-solving decisions.

TRIZ, which stands for “The Theory of Inventive Problem Solving,” is a problem-solving, analysis, and forecasting methodology.

It focuses on finding contradictions inherent in a scenario. Then, you work toward eliminating the contraditions through finding innovative solutions.

So, when you’re tackling a problem, TRIZ provides a disciplined, systematic approach that aims for ideal solutions and not just acceptable ones. Using TRIZ, you can leverage patterns of problem-solving that have proven effective in different cases, pivoting them to solve the problem at hand.

38. A3 Problem Solving

A3 Problem Solving, derived from Lean Management, is a structured method that uses a single sheet of A3-sized paper to document knowledge from a problem-solving process.

Named after the international paper size standard of A3 (or 11-inch by 17-inch paper), it succinctly records all key details of the problem-solving process from problem description to the root cause and corrective actions.

Used in problem-solving, this provides a straightforward and logical structure for addressing the problem, facilitating communication between team members, ensuring all critical details are included, and providing a record of decisions made.

39. Scenario Analysis

Scenario Analysis is all about predicting different possible future events depending upon your decision.

To do this, you look at each course of action and try to identify the most likely outcomes or scenarios down the track if you take that course of action.

This technique helps forecast the impacts of various strategies, playing each out to their (logical or potential) end. It’s a good strategy for project managers who need to keep a firm eye on the horizon at all times.

When solving problems, Scenario Analysis assists in preparing for uncertainties, making sure your solution remains viable, regardless of changes in circumstances.

How to Answer “Demonstrate Problem-Solving Skills” in an Interview

When asked to demonstrate your problem-solving skills in an interview, the STAR method often proves useful. STAR stands for Situation, Task, Action, and Result.

Situation: Begin by describing a specific circumstance or challenge you encountered. Make sure to provide enough detail to allow the interviewer a clear understanding. You should select an event that adequately showcases your problem-solving abilities.

For instance, “In my previous role as a project manager, we faced a significant issue when our key supplier abruptly went out of business.”

Task: Explain what your responsibilities were in that situation. This serves to provide context, allowing the interviewer to understand your role and the expectations placed upon you.

For instance, “It was my task to ensure the project remained on track despite this setback. Alternative suppliers needed to be found without sacrificing quality or significantly increasing costs.”

Action: Describe the steps you took to manage the problem. Highlight your problem-solving process. Mention any creative approaches or techniques that you used.

For instance, “I conducted thorough research to identify potential new suppliers. After creating a shortlist, I initiated contact, negotiated terms, assessed samples for quality and made a selection. I also worked closely with the team to re-adjust the project timeline.”

Result: Share the outcomes of your actions. How did the situation end? Did your actions lead to success? It’s particularly effective if you can quantify these results.

For instance, “As a result of my active problem solving, we were able to secure a new supplier whose costs were actually 10% cheaper and whose quality was comparable. We adjusted the project plan and managed to complete the project just two weeks later than originally planned, despite the major vendor setback.”

Remember, when you’re explaining your problem-solving skills to an interviewer, what they’re really interested in is your approach to handling difficulties, your creativity and persistence in seeking a resolution, and your ability to carry your solution through to fruition. Tailoring your story to highlight these aspects will help exemplify your problem-solving prowess.

Go Deeper: STAR Interview Method Examples

Benefits of Problem-Solving

Problem-solving is beneficial for the following reasons (among others):

• It can help you to overcome challenges, roadblocks, and bottlenecks in your life.
• It can save a company money.
• It can help you to achieve clarity in your thinking.
• It can make procedures more efficient and save time.
• It can strengthen your decision-making capacities.
• It can lead to better risk management.

Whether for a job interview or school, problem-solving helps you to become a better thinking, solve your problems more effectively, and achieve your goals. Build up your problem-solving frameworks (I presented over 40 in this piece for you!) and work on applying them in real-life situations.

Chris Drew (PhD)

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

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ Social-Emotional Learning (Definition, Examples, Pros & Cons)
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ What is Educational Psychology?
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ What is IQ? (Intelligence Quotient)
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 5 Top Tips for Succeeding at University

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35 problem-solving techniques and methods for solving complex problems

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

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

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

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

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

Let’s get started! 

How do you identify problems?

How do you identify the right solution.

• Tips for more effective problem-solving

Complete problem-solving methods

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

Problem-solving warm-up activities

Closing activities for a problem-solving process.

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

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

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

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

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

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

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

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

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

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

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

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

Tips for more effective problem solving

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

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

Clearly define the problem

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

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

Don’t jump to conclusions

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

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

Try different approaches  

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

Don’t take it personally 

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

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

Get the right people in the room

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

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

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

Document everything

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

Bring a facilitator 

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

Develop your problem-solving skills

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

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

Design a great agenda

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

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

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

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

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

Design Sprint 2.0

• Open Space Technology

1. Six Thinking Hats

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

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

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

2. Lightning Decision Jam

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

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

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

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

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

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

3. Problem Definition Process

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

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

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

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

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

4. The 5 Whys 

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

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

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

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

5. World Cafe

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

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

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

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

6. Discovery & Action Dialogue (DAD)

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

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

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

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

7. Design Sprint 2.0

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

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

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

8. Open space technology

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

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

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

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

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

Techniques to identify and analyze problems

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

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

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

Let’s take a look!

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

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

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

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

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

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

10. The Creativity Dice

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

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

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

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

11. Fishbone Analysis

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

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

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

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

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

12. Problem Tree 

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

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

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

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

13. SWOT Analysis

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

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

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

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

14. Agreement-Certainty Matrix

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

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

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

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

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

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

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

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

16. Speed Boat

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

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

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

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

17. The Journalistic Six

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

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

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

18. LEGO Challenge

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

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

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

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

19. What, So What, Now What?

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

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

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

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

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

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

20. Journalists  

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

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

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

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

Problem-solving techniques for developing solutions 

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

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

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

21. Mindspin  

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

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

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

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

22. Improved Solutions

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

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

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

23. Four Step Sketch

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

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

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

24. 15% Solutions

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

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

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

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

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

25. How-Now-Wow Matrix

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

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

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

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

26. Impact and Effort Matrix

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

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

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

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

27. Dotmocracy

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

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

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

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

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

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

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

28. Check-in / Check-out

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

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

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

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

29. Doodling Together  

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

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

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

30. Show and Tell

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

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

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

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

31. Constellations

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

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

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

32. Draw a Tree

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

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

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

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

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

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

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

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

How do I conclude a problem-solving process?

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

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

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

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

33. One Breath Feedback

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

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

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

34. Who What When Matrix 

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

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

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

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

35. Response cards

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

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

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

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

Save time and effort discovering the right solutions

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

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

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

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

Over to you

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

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

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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Problem-Solving Strategies and Obstacles

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

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

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• Application
• Improvement

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

What Is Problem-Solving?

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

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

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

The problem-solving process involves:

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

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

Problem-Solving Mental Processes

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

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

Problem-Solving Strategies

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

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

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

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

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

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

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

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

Trial and Error

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

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

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

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

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

How to Apply Problem-Solving Strategies in Real Life

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

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

Obstacles to Problem-Solving

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

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

How to Improve Your Problem-Solving Skills

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

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

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

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

Top 20 Problem Solving Interview Questions (Example Answers Included)

Mike Simpson 0 Comments

By Mike Simpson

When candidates prepare for interviews, they usually focus on highlighting their leadership, communication, teamwork, and similar crucial soft skills . However, not everyone gets ready for problem-solving interview questions. And that can be a big mistake.

Problem-solving is relevant to nearly any job on the planet. Yes, it’s more prevalent in certain industries, but it’s helpful almost everywhere.

Regardless of the role you want to land, you may be asked to provide problem-solving examples or describe how you would deal with specific situations. That’s why being ready to showcase your problem-solving skills is so vital.

If you aren’t sure who to tackle problem-solving questions, don’t worry, we have your back. Come with us as we explore this exciting part of the interview process, as well as some problem-solving interview questions and example answers.

What Is Problem-Solving?

When you’re trying to land a position, there’s a good chance you’ll face some problem-solving interview questions. But what exactly is problem-solving? And why is it so important to hiring managers?

Well, the good folks at Merriam-Webster define problem-solving as “the process or act of finding a solution to a problem.” While that may seem like common sense, there’s a critical part to that definition that should catch your eye.

What part is that? The word “process.”

In the end, problem-solving is an activity. It’s your ability to take appropriate steps to find answers, determine how to proceed, or otherwise overcome the challenge.

Being great at it usually means having a range of helpful problem-solving skills and traits. Research, diligence, patience, attention-to-detail , collaboration… they can all play a role. So can analytical thinking , creativity, and open-mindedness.

But why do hiring managers worry about your problem-solving skills? Well, mainly, because every job comes with its fair share of problems.

While problem-solving is relevant to scientific, technical, legal, medical, and a whole slew of other careers. It helps you overcome challenges and deal with the unexpected. It plays a role in troubleshooting and innovation. That’s why it matters to hiring managers.

How to Answer Problem-Solving Interview Questions

Okay, before we get to our examples, let’s take a quick second to talk about strategy. Knowing how to answer problem-solving interview questions is crucial. Why? Because the hiring manager might ask you something that you don’t anticipate.

Problem-solving interview questions are all about seeing how you think. As a result, they can be a bit… unconventional.

These aren’t your run-of-the-mill job interview questions . Instead, they are tricky behavioral interview questions . After all, the goal is to find out how you approach problem-solving, so most are going to feature scenarios, brainteasers, or something similar.

So, having a great strategy means knowing how to deal with behavioral questions. Luckily, there are a couple of tools that can help.

First, when it comes to the classic approach to behavioral interview questions, look no further than the STAR Method . With the STAR method, you learn how to turn your answers into captivating stories. This makes your responses tons more engaging, ensuring you keep the hiring manager’s attention from beginning to end.

Now, should you stop with the STAR Method? Of course not. If you want to take your answers to the next level, spend some time with the Tailoring Method , too.

With the Tailoring Method, it’s all about relevance. So, if you get a chance to choose an example that demonstrates your problem-solving skills, this is really the way to go.

We also wanted to let you know that we created an amazing free cheat sheet that will give you word-for-word answers for some of the toughest interview questions you are going to face in your upcoming interview. After all, hiring managers will often ask you more generalized interview questions!

Click below to get your free PDF now:

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Top 3 Problem-Solving-Based Interview Questions

Alright, here is what you’ve been waiting for: the problem-solving questions and sample answers.

While many questions in this category are job-specific, these tend to apply to nearly any job. That means there’s a good chance you’ll come across them at some point in your career, making them a great starting point when you’re practicing for an interview.

So, let’s dive in, shall we? Here’s a look at the top three problem-solving interview questions and example responses.

1. Can you tell me about a time when you had to solve a challenging problem?

In the land of problem-solving questions, this one might be your best-case scenario. It lets you choose your own problem-solving examples to highlight, putting you in complete control.

When you choose an example, go with one that is relevant to what you’ll face in the role. The closer the match, the better the answer is in the eyes of the hiring manager.

EXAMPLE ANSWER:

“While working as a mobile telecom support specialist for a large organization, we had to transition our MDM service from one vendor to another within 45 days. This personally physically handling 500 devices within the agency. Devices had to be gathered from the headquarters and satellite offices, which were located all across the state, something that was challenging even without the tight deadline. I approached the situation by identifying the location assignment of all personnel within the organization, enabling me to estimate transit times for receiving the devices. Next, I timed out how many devices I could personally update in a day. Together, this allowed me to create a general timeline. After that, I coordinated with each location, both expressing the urgency of adhering to deadlines and scheduling bulk shipping options. While there were occasional bouts of resistance, I worked with location leaders to calm concerns and facilitate action. While performing all of the updates was daunting, my approach to organizing the event made it a success. Ultimately, the entire transition was finished five days before the deadline, exceeding the expectations of many.”

2. Describe a time where you made a mistake. What did you do to fix it?

While this might not look like it’s based on problem-solving on the surface, it actually is. When you make a mistake, it creates a challenge, one you have to work your way through. At a minimum, it’s an opportunity to highlight problem-solving skills, even if you don’t address the topic directly.

When you choose an example, you want to go with a situation where the end was positive. However, the issue still has to be significant, causing something negative to happen in the moment that you, ideally, overcame.

“When I first began in a supervisory role, I had trouble setting down my individual contributor hat. I tried to keep up with my past duties while also taking on the responsibilities of my new role. As a result, I began rushing and introduced an error into the code of the software my team was updating. The error led to a memory leak. We became aware of the issue when the performance was hindered, though we didn’t immediately know the cause. I dove back into the code, reviewing recent changes, and, ultimately, determined the issue was a mistake on my end. When I made that discovery, I took several steps. First, I let my team know that the error was mine and let them know its nature. Second, I worked with my team to correct the issue, resolving the memory leak. Finally, I took this as a lesson about delegation. I began assigning work to my team more effectively, a move that allowed me to excel as a manager and help them thrive as contributors. It was a crucial learning moment, one that I have valued every day since.”

3. If you identify a potential risk in a project, what steps do you take to prevent it?

Yes, this is also a problem-solving question. The difference is, with this one, it’s not about fixing an issue; it’s about stopping it from happening. Still, you use problem-solving skills along the way, so it falls in this question category.

If you can, use an example of a moment when you mitigated risk in the past. If you haven’t had that opportunity, approach it theoretically, discussing the steps you would take to prevent an issue from developing.

“If I identify a potential risk in a project, my first step is to assess the various factors that could lead to a poor outcome. Prevention requires analysis. Ensuring I fully understand what can trigger the undesired event creates the right foundation, allowing me to figure out how to reduce the likelihood of those events occurring. Once I have the right level of understanding, I come up with a mitigation plan. Exactly what this includes varies depending on the nature of the issue, though it usually involves various steps and checks designed to monitor the project as it progresses to spot paths that may make the problem more likely to happen. I find this approach effective as it combines knowledge and ongoing vigilance. That way, if the project begins to head into risky territory, I can correct its trajectory.”

17 More Problem-Solving-Based Interview Questions

In the world of problem-solving questions, some apply to a wide range of jobs, while others are more niche. For example, customer service reps and IT helpdesk professionals both encounter challenges, but not usually the same kind.

As a result, some of the questions in this list may be more relevant to certain careers than others. However, they all give you insights into what this kind of question looks like, making them worth reviewing.

Here are 17 more problem-solving interview questions you might face off against during your job search:

• How would you describe your problem-solving skills?
• Can you tell me about a time when you had to use creativity to deal with an obstacle?
• Describe a time when you discovered an unmet customer need while assisting a customer and found a way to meet it.
• If you were faced with an upset customer, how would you diffuse the situation?
• Tell me about a time when you had to troubleshoot a complex issue.
• Imagine you were overseeing a project and needed a particular item. You have two choices of vendors: one that can deliver on time but would be over budget, and one that’s under budget but would deliver one week later than you need it. How do you figure out which approach to use?
• Your manager wants to upgrade a tool you regularly use for your job and wants your recommendation. How do you formulate one?
• A supplier has said that an item you need for a project isn’t going to be delivered as scheduled, something that would cause your project to fall behind schedule. What do you do to try and keep the timeline on target?
• Can you share an example of a moment where you encountered a unique problem you and your colleagues had never seen before? How did you figure out what to do?
• Imagine you were scheduled to give a presentation with a colleague, and your colleague called in sick right before it was set to begin. What would you do?
• If you are given two urgent tasks from different members of the leadership team, both with the same tight deadline, how do you choose which to tackle first?
• Tell me about a time you and a colleague didn’t see eye-to-eye. How did you decide what to do?
• Describe your troubleshooting process.
• Tell me about a time where there was a problem that you weren’t able to solve. What happened?
• In your opening, what skills or traits make a person an exceptional problem-solver?
• When you face a problem that requires action, do you usually jump in or take a moment to carefully assess the situation?
• When you encounter a new problem you’ve never seen before, what is the first step that you take?

Putting It All Together

At this point, you should have a solid idea of how to approach problem-solving interview questions. Use the tips above to your advantage. That way, you can thrive during your next interview.

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Co-Founder and CEO of TheInterviewGuys.com. Mike is a job interview and career expert and the head writer at TheInterviewGuys.com. His advice and insights have been shared and featured by publications such as Forbes , Entrepreneur , CNBC and more as well as educational institutions such as the University of Michigan , Penn State , Northeastern and others. Learn more about The Interview Guys on our About Us page .

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50 Problem-Solving and Critical Thinking Examples

Critical thinking and problem solving are essential skills for success in the 21st century. Critical thinking is the ability to analyze information, evaluate evidence, and draw logical conclusions. Problem solving is the ability to apply critical thinking to find effective solutions to various challenges. Both skills require creativity, curiosity, and persistence. Developing critical thinking and problem solving skills can help students improve their academic performance, enhance their career prospects, and become more informed and engaged citizens.

Sanju Pradeepa

In today’s complex and fast-paced world, the ability to think critically and solve problems effectively has become a vital skill for success in all areas of life. Whether it’s navigating professional challenges, making sound decisions, or finding innovative solutions, critical thinking and problem-solving are key to overcoming obstacles and achieving desired outcomes. In this blog post, we will explore problem-solving and critical thinking examples.

Table of Contents

Developing the skills needed for critical thinking and problem solving.

It is not enough to simply recognize an issue; we must use the right tools and techniques to address it. To do this, we must learn how to define and identify the problem or task at hand, gather relevant information from reliable sources, analyze and compare data to draw conclusions, make logical connections between different ideas, generate a solution or action plan, and make a recommendation.

The first step in developing these skills is understanding what the problem or task is that needs to be addressed. This requires careful consideration of all available information in order to form an accurate picture of what needs to be done. Once the issue has been identified, gathering reliable sources of data can help further your understanding of it. Sources could include interviews with customers or stakeholders, surveys, industry reports, and analysis of customer feedback.

After collecting relevant information from reliable sources, it’s important to analyze and compare the data in order to draw meaningful conclusions about the situation at hand. This helps us better understand our options for addressing an issue by providing context for decision-making. Once you have analyzed the data you collected, making logical connections between different ideas can help you form a more complete picture of the situation and inform your potential solutions.

Once you have analyzed your options for addressing an issue based on all available data points, it’s time to generate a solution or action plan that takes into account considerations such as cost-effectiveness and feasibility. It’s also important to consider the risk factors associated with any proposed solutions in order to ensure that they are responsible before moving forward with implementation. Finally, once all the analysis has been completed, it is time to make a recommendation based on your findings, which should take into account any objectives set out by stakeholders at the beginning of this process as well as any other pertinent factors discovered throughout the analysis stage.

By following these steps carefully when faced with complex issues, one can effectively use critical thinking and problem-solving skills in order to achieve desired outcomes more efficiently than would otherwise be possible without them, while also taking responsibility for decisions made along the way.

What Does Critical Thinking Involve: 5 Essential Skill

Problem-solving and critical thinking examples.

Problem-solving and critical thinking are key skills that are highly valued in any professional setting. These skills enable individuals to analyze complex situations, make informed decisions, and find innovative solutions. Here, we present 25 examples of problem-solving and critical thinking. problem-solving scenarios to help you cultivate and enhance these skills.

Ethical dilemma: A company faces a situation where a client asks for a product that does not meet quality standards. The team must decide how to address the client’s request without compromising the company’s credibility or values.

Brainstorming session: A team needs to come up with new ideas for a marketing campaign targeting a specific demographic. Through an organized brainstorming session, they explore various approaches and analyze their potential impact.

Troubleshooting technical issues : An IT professional receives a ticket indicating a network outage. They analyze the issue, assess potential causes (hardware, software, or connectivity), and solve the problem efficiently.

Negotiation : During contract negotiations, representatives from two companies must find common ground to strike a mutually beneficial agreement, considering the needs and limitations of both parties.

Project management: A project manager identifies potential risks and develops contingency plans to address unforeseen obstacles, ensuring the project stays on track.

Decision-making under pressure: In a high-stakes situation, a medical professional must make a critical decision regarding a patient’s treatment, weighing all available information and considering potential risks.

Conflict resolution: A team encounters conflicts due to differing opinions or approaches. The team leader facilitates a discussion to reach a consensus while considering everyone’s perspectives.

Data analysis: A data scientist is presented with a large dataset and is tasked with extracting valuable insights. They apply analytical techniques to identify trends, correlations, and patterns that can inform decision-making.

Customer service: A customer service representative encounters a challenging customer complaint and must employ active listening and problem-solving skills to address the issue and provide a satisfactory resolution.

Market research : A business seeks to expand into a new market. They conduct thorough market research, analyzing consumer behavior, competitor strategies, and economic factors to make informed market-entry decisions.

Creative problem-solvin g: An engineer faces a design challenge and must think outside the box to come up with a unique and innovative solution that meets project requirements.

Change management: During a company-wide transition, managers must effectively communicate the change, address employees’ concerns, and facilitate a smooth transition process.

Crisis management: When a company faces a public relations crisis, effective critical thinking is necessary to analyze the situation, develop a response strategy, and minimize potential damage to the company’s reputation.

Cost optimization : A financial analyst identifies areas where expenses can be reduced while maintaining operational efficiency, presenting recommendations for cost savings.

Time management : An employee has multiple deadlines to meet. They assess the priority of each task, develop a plan, and allocate time accordingly to achieve optimal productivity.

Quality control: A production manager detects an increase in product defects and investigates the root causes, implementing corrective actions to enhance product quality.

Strategic planning: An executive team engages in strategic planning to define long-term goals, assess market trends, and identify growth opportunities.

Cross-functional collaboration: Multiple teams with different areas of expertise must collaborate to develop a comprehensive solution, combining their knowledge and skills.

Training and development : A manager identifies skill gaps in their team and designs training programs to enhance critical thinking, problem-solving, and decision-making abilities.

Risk assessment : A risk management professional evaluates potential risks associated with a new business venture, weighing their potential impact and developing strategies to mitigate them.

Continuous improvement: An operations manager analyzes existing processes, identifies inefficiencies, and introduces improvements to enhance productivity and customer satisfaction.

Customer needs analysis: A product development team conducts extensive research to understand customer needs and preferences, ensuring that the resulting product meets those requirements.

Crisis decision-making: A team dealing with a crisis must think quickly, assess the situation, and make timely decisions with limited information.

Marketing campaign analysis : A marketing team evaluates the success of a recent campaign, analyzing key performance indicators to understand its impact on sales and customer engagement.

Constructive feedback: A supervisor provides feedback to an employee, highlighting areas for improvement and offering constructive suggestions for growth.

Conflict resolution in a team project: Team members engaged in a project have conflicting ideas on the approach. They must engage in open dialogue, actively listen to each other’s perspectives, and reach a compromise that aligns with the project’s goals.

Crisis response in a natural disaster: Emergency responders must think critically and swiftly in responding to a natural disaster, coordinating rescue efforts, allocating resources effectively, and prioritizing the needs of affected individuals.

Product innovation : A product development team conducts market research, studies consumer trends, and uses critical thinking to create innovative products that address unmet customer needs.

Supply chain optimization: A logistics manager analyzes the supply chain to identify areas for efficiency improvement, such as reducing transportation costs, improving inventory management, or streamlining order fulfillment processes.

Business strategy formulation: A business executive assesses market dynamics, the competitive landscape, and internal capabilities to develop a robust business strategy that ensures sustainable growth and competitiveness.

Crisis communication: In the face of a public relations crisis, an organization’s spokesperson must think critically to develop and deliver a transparent, authentic, and effective communication strategy to rebuild trust and manage reputation.

Social problem-solving: A group of volunteers addresses a specific social issue, such as poverty or homelessness, by critically examining its root causes, collaborating with stakeholders, and implementing sustainable solutions for the affected population.

Problem-Solving Mindset: How to Achieve It (15 Ways)

Risk assessment in investment decision-making: An investment analyst evaluates various investment opportunities, conducting risk assessments based on market trends, financial indicators, and potential regulatory changes to make informed investment recommendations.

Environmental sustainability: An environmental scientist analyzes the impact of industrial processes on the environment, develops strategies to mitigate risks, and promotes sustainable practices within organizations and communities.

Adaptation to technological advancements : In a rapidly evolving technological landscape, professionals need critical thinking skills to adapt to new tools, software, and systems, ensuring they can effectively leverage these advancements to enhance productivity and efficiency.

Productivity improvement: An operations manager leverages critical thinking to identify productivity bottlenecks within a workflow and implement process improvements to optimize resource utilization, minimize waste, and increase overall efficiency.

Cost-benefit analysis: An organization considering a major investment or expansion opportunity conducts a thorough cost-benefit analysis, weighing potential costs against expected benefits to make an informed decision.

Human resources management : HR professionals utilize critical thinking to assess job applicants, identify skill gaps within the organization, and design training and development programs to enhance the workforce’s capabilities.

Root cause analysis: In response to a recurring problem or inefficiency, professionals apply critical thinking to identify the root cause of the issue, develop remedial actions, and prevent future occurrences.

Leadership development: Aspiring leaders undergo critical thinking exercises to enhance their decision-making abilities, develop strategic thinking skills, and foster a culture of innovation within their teams.

Brand positioning : Marketers conduct comprehensive market research and consumer behavior analysis to strategically position a brand, differentiating it from competitors and appealing to target audiences effectively.

Resource allocation: Non-profit organizations distribute limited resources efficiently, critically evaluating project proposals, considering social impact, and allocating resources to initiatives that align with their mission.

Innovating in a mature market: A company operating in a mature market seeks to innovate to maintain a competitive edge. They cultivate critical thinking skills to identify gaps, anticipate changing customer needs, and develop new strategies, products, or services accordingly.

Analyzing financial statements : Financial analysts critically assess financial statements, analyze key performance indicators, and derive insights to support financial decision-making, such as investment evaluations or budget planning.

Crisis intervention : Mental health professionals employ critical thinking and problem-solving to assess crises faced by individuals or communities, develop intervention plans, and provide support during challenging times.

Data privacy and cybersecurity : IT professionals critically evaluate existing cybersecurity measures, identify vulnerabilities, and develop strategies to protect sensitive data from threats, ensuring compliance with privacy regulations.

Process improvement : Professionals in manufacturing or service industries critically evaluate existing processes, identify inefficiencies, and implement improvements to optimize efficiency, quality, and customer satisfaction.

Multi-channel marketing strategy : Marketers employ critical thinking to design and execute effective marketing campaigns across various channels such as social media, web, print, and television, ensuring a cohesive brand experience for customers.

Peer review: Researchers critically analyze and review the work of their peers, providing constructive feedback and ensuring the accuracy, validity, and reliability of scientific studies.

Project coordination : A project manager must coordinate multiple teams and resources to ensure seamless collaboration, identify potential bottlenecks, and find solutions to keep the project on schedule.  

These examples highlight the various contexts in which problem-solving and critical-thinking skills are necessary for success. By understanding and practicing these skills, individuals can enhance their ability to navigate challenges and make sound decisions in both personal and professional endeavors.

Conclusion:

Critical thinking and problem-solving are indispensable skills that empower individuals to overcome challenges, make sound decisions, and find innovative solutions. By honing these skills, one can navigate through the complexities of modern life and achieve success in both personal and professional endeavors. Embrace the power of critical thinking and problem-solving, and unlock the door to endless possibilities and growth.

• Problem solving From Wikipedia, the free encyclopedia
• Critical thinking From Wikipedia, the free encyclopedia
• The Importance of Critical Thinking and Problem Solving Skills for Students (5 Minutes)

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Culture Development

Workplace problem-solving examples: real scenarios, practical solutions.

• March 11, 2024

In today’s fast-paced and ever-changing work environment, problems are inevitable. From conflicts among employees to high levels of stress, workplace problems can significantly impact productivity and overall well-being. However, by developing the art of problem-solving and implementing practical solutions, organizations can effectively tackle these challenges and foster a positive work culture. In this article, we will delve into various workplace problem scenarios and explore strategies for resolution. By understanding common workplace problems and acquiring essential problem-solving skills, individuals and organizations can navigate these challenges with confidence and success.

Understanding Workplace Problems

Before we can effectively solve workplace problems , it is essential to gain a clear understanding of the issues at hand. Identifying common workplace problems is the first step toward finding practical solutions. By recognizing these challenges, organizations can develop targeted strategies and initiatives to address them.

Identifying Common Workplace Problems

One of the most common workplace problems is conflict. Whether it stems from differences in opinions, miscommunication, or personality clashes, conflict can disrupt collaboration and hinder productivity. It is important to note that conflict is a natural part of any workplace, as individuals with different backgrounds and perspectives come together to work towards a common goal. However, when conflict is not managed effectively, it can escalate and create a toxic work environment.

In addition to conflict, workplace stress and burnout pose significant challenges. High workloads, tight deadlines, and a lack of work-life balance can all contribute to employee stress and dissatisfaction. When employees are overwhelmed and exhausted, their performance and overall well-being are compromised. This not only affects the individuals directly, but it also has a ripple effect on the entire organization.

Another common workplace problem is poor communication. Ineffective communication can lead to misunderstandings, delays, and errors. It can also create a sense of confusion and frustration among employees. Clear and open communication is vital for successful collaboration and the smooth functioning of any organization.

The Impact of Workplace Problems on Productivity

Workplace problems can have a detrimental effect on productivity levels. When conflicts are left unresolved, they can create a tense work environment, leading to decreased employee motivation and engagement. The negative energy generated by unresolved conflicts can spread throughout the organization, affecting team dynamics and overall performance.

Similarly, high levels of stress and burnout can result in decreased productivity, as individuals may struggle to focus and perform optimally. When employees are constantly under pressure and overwhelmed, their ability to think creatively and problem-solve diminishes. This can lead to a decline in the quality of work produced and an increase in errors and inefficiencies.

Poor communication also hampers productivity. When information is not effectively shared or understood, it can lead to misunderstandings, delays, and rework. This not only wastes time and resources but also creates frustration and demotivation among employees.

Furthermore, workplace problems can negatively impact employee morale and job satisfaction. When individuals are constantly dealing with conflicts, stress, and poor communication, their overall job satisfaction and engagement suffer. This can result in higher turnover rates, as employees seek a healthier and more supportive work environment.

In conclusion, workplace problems such as conflict, stress, burnout, and poor communication can significantly hinder productivity and employee well-being. Organizations must address these issues promptly and proactively to create a positive and productive work atmosphere. By fostering open communication, providing support for stress management, and promoting conflict resolution strategies, organizations can create a work environment that encourages collaboration, innovation, and employee satisfaction.

The Art of Problem Solving in the Workplace

Now that we have a clear understanding of workplace problems, let’s explore the essential skills necessary for effective problem-solving in the workplace. By developing these skills and adopting a proactive approach, individuals can tackle problems head-on and find practical solutions.

Problem-solving in the workplace is a complex and multifaceted skill that requires a combination of analytical thinking, creativity, and effective communication. It goes beyond simply identifying problems and extends to finding innovative solutions that address the root causes.

Essential Problem-Solving Skills for the Workplace

To effectively solve workplace problems, individuals should possess a range of skills. These include strong analytical and critical thinking abilities, excellent communication and interpersonal skills, the ability to collaborate and work well in a team, and the capacity to adapt to change. By honing these skills, individuals can approach workplace problems with confidence and creativity.

Analytical and critical thinking skills are essential for problem-solving in the workplace. They involve the ability to gather and analyze relevant information, identify patterns and trends, and make logical connections. These skills enable individuals to break down complex problems into manageable components and develop effective strategies to solve them.

Effective communication and interpersonal skills are also crucial for problem-solving in the workplace. These skills enable individuals to clearly articulate their thoughts and ideas, actively listen to others, and collaborate effectively with colleagues. By fostering open and honest communication channels, individuals can better understand the root causes of problems and work towards finding practical solutions.

Collaboration and teamwork are essential for problem-solving in the workplace. By working together, individuals can leverage their diverse skills, knowledge, and perspectives to generate innovative solutions. Collaboration fosters a supportive and inclusive environment where everyone’s ideas are valued, leading to more effective problem-solving outcomes.

The ability to adapt to change is another important skill for problem-solving in the workplace. In today’s fast-paced and dynamic work environment, problems often arise due to changes in technology, processes, or market conditions. Individuals who can embrace change and adapt quickly are better equipped to find solutions that address the evolving needs of the organization.

The Role of Communication in Problem Solving

Communication is a key component of effective problem-solving in the workplace. By fostering open and honest communication channels, individuals can better understand the root causes of problems and work towards finding practical solutions. Active listening, clear and concise articulation of thoughts and ideas, and the ability to empathize are all valuable communication skills that facilitate problem-solving.

Active listening involves fully engaging with the speaker, paying attention to both verbal and non-verbal cues, and seeking clarification when necessary. By actively listening, individuals can gain a deeper understanding of the problem at hand and the perspectives of others involved. This understanding is crucial for developing comprehensive and effective solutions.

Clear and concise articulation of thoughts and ideas is essential for effective problem-solving communication. By expressing oneself clearly, individuals can ensure that their ideas are understood by others. This clarity helps to avoid misunderstandings and promotes effective collaboration.

Empathy is a valuable communication skill that plays a significant role in problem-solving. By putting oneself in the shoes of others and understanding their emotions and perspectives, individuals can build trust and rapport. This empathetic connection fosters a supportive and collaborative environment where everyone feels valued and motivated to contribute to finding solutions.

In conclusion, problem-solving in the workplace requires a combination of essential skills such as analytical thinking, effective communication, collaboration, and adaptability. By honing these skills and fostering open communication channels, individuals can approach workplace problems with confidence and creativity, leading to practical and innovative solutions.

Real Scenarios of Workplace Problems

Now, let’s explore some real scenarios of workplace problems and delve into strategies for resolution. By examining these practical examples, individuals can develop a deeper understanding of how to approach and solve workplace problems.

Conflict Resolution in the Workplace

Imagine a scenario where two team members have conflicting ideas on how to approach a project. The disagreement becomes heated, leading to a tense work environment. To resolve this conflict, it is crucial to encourage open dialogue between the team members. Facilitating a calm and respectful conversation can help uncover underlying concerns and find common ground. Collaboration and compromise are key in reaching a resolution that satisfies all parties involved.

In this particular scenario, let’s dive deeper into the dynamics between the team members. One team member, let’s call her Sarah, strongly believes that a more conservative and traditional approach is necessary for the project’s success. On the other hand, her colleague, John, advocates for a more innovative and out-of-the-box strategy. The clash between their perspectives arises from their different backgrounds and experiences.

As the conflict escalates, it is essential for a neutral party, such as a team leader or a mediator, to step in and facilitate the conversation. This person should create a safe space for both Sarah and John to express their ideas and concerns without fear of judgment or retribution. By actively listening to each other, they can gain a better understanding of the underlying motivations behind their respective approaches.

During the conversation, it may become apparent that Sarah’s conservative approach stems from a fear of taking risks and a desire for stability. On the other hand, John’s innovative mindset is driven by a passion for pushing boundaries and finding creative solutions. Recognizing these underlying motivations can help foster empathy and create a foundation for collaboration.

As the dialogue progresses, Sarah and John can begin to identify areas of overlap and potential compromise. They may realize that while Sarah’s conservative approach provides stability, John’s innovative ideas can inject fresh perspectives into the project. By combining their strengths and finding a middle ground, they can develop a hybrid strategy that incorporates both stability and innovation.

Ultimately, conflict resolution in the workplace requires effective communication, active listening, empathy, and a willingness to find common ground. By addressing conflicts head-on and fostering a collaborative environment, teams can overcome challenges and achieve their goals.

Dealing with Workplace Stress and Burnout

Workplace stress and burnout can be debilitating for individuals and organizations alike. In this scenario, an employee is consistently overwhelmed by their workload and experiencing signs of burnout. To address this issue, organizations should promote a healthy work-life balance and provide resources to manage stress effectively. Encouraging employees to take breaks, providing access to mental health support, and fostering a supportive work culture are all practical solutions to alleviate workplace stress.

In this particular scenario, let’s imagine that the employee facing stress and burnout is named Alex. Alex has been working long hours, often sacrificing personal time and rest to meet tight deadlines and demanding expectations. As a result, Alex is experiencing physical and mental exhaustion, reduced productivity, and a sense of detachment from work.

Recognizing the signs of burnout, Alex’s organization takes proactive measures to address the issue. They understand that employee well-being is crucial for maintaining a healthy and productive workforce. To promote a healthy work-life balance, the organization encourages employees to take regular breaks and prioritize self-care. They emphasize the importance of disconnecting from work during non-working hours and encourage employees to engage in activities that promote relaxation and rejuvenation.

Additionally, the organization provides access to mental health support services, such as counseling or therapy sessions. They recognize that stress and burnout can have a significant impact on an individual’s mental well-being and offer resources to help employees manage their stress effectively. By destigmatizing mental health and providing confidential support, the organization creates an environment where employees feel comfortable seeking help when needed.

Furthermore, the organization fosters a supportive work culture by promoting open communication and empathy. They encourage managers and colleagues to check in with each other regularly, offering support and understanding. Team members are encouraged to collaborate and share the workload, ensuring that no one person is overwhelmed with excessive responsibilities.

By implementing these strategies, Alex’s organization aims to alleviate workplace stress and prevent burnout. They understand that a healthy and balanced workforce is more likely to be engaged, productive, and satisfied. Through a combination of promoting work-life balance, providing mental health support, and fostering a supportive work culture, organizations can effectively address workplace stress and create an environment conducive to employee well-being.

Practical Solutions to Workplace Problems

Now that we have explored real scenarios, let’s discuss practical solutions that organizations can implement to address workplace problems. By adopting proactive strategies and establishing effective policies, organizations can create a positive work environment conducive to problem-solving and productivity.

Implementing Effective Policies for Problem Resolution

Organizations should have clear and well-defined policies in place to address workplace problems. These policies should outline procedures for conflict resolution, channels for reporting problems, and accountability measures. By ensuring that employees are aware of these policies and have easy access to them, organizations can facilitate problem-solving and prevent issues from escalating.

Promoting a Positive Workplace Culture

A positive workplace culture is vital for problem-solving. By fostering an environment of respect, collaboration, and open communication, organizations can create a space where individuals feel empowered to address and solve problems. Encouraging teamwork, recognizing and appreciating employees’ contributions, and promoting a healthy work-life balance are all ways to cultivate a positive workplace culture.

The Role of Leadership in Problem Solving

Leadership plays a crucial role in facilitating effective problem-solving within organizations. Different leadership styles can impact how problems are approached and resolved.

Leadership Styles and Their Impact on Problem-Solving

Leaders who adopt an autocratic leadership style may make decisions independently, potentially leaving their team members feeling excluded and undervalued. On the other hand, leaders who adopt a democratic leadership style involve their team members in the problem-solving process, fostering a sense of ownership and empowerment. By encouraging employee participation, organizations can leverage the diverse perspectives and expertise of their workforce to find innovative solutions to workplace problems.

Encouraging Employee Participation in Problem Solving

To harness the collective problem-solving abilities of an organization, it is crucial to encourage employee participation. Leaders can create opportunities for employees to contribute their ideas and perspectives through brainstorming sessions, team meetings, and collaborative projects. By valuing employee input and involving them in decision-making processes, organizations can foster a culture of inclusivity and drive innovative problem-solving efforts.

In today’s dynamic work environment, workplace problems are unavoidable. However, by understanding common workplace problems, developing essential problem-solving skills, and implementing practical solutions, individuals and organizations can navigate these challenges effectively. By fostering a positive work culture, implementing effective policies, and encouraging employee participation, organizations can create an environment conducive to problem-solving and productivity. With proactive problem-solving strategies in place, organizations can thrive and overcome obstacles, ensuring long-term success and growth.

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How Good Is Your Problem Solving?

Use a systematic approach..

By the Mind Tools Content Team

Good problem solving skills are fundamentally important if you're going to be successful in your career.

But problems are something that we don't particularly like.

They're time-consuming.

They muscle their way into already packed schedules.

They force us to think about an uncertain future.

And they never seem to go away!

That's why, when faced with problems, most of us try to eliminate them as quickly as possible. But have you ever chosen the easiest or most obvious solution – and then realized that you have entirely missed a much better solution? Or have you found yourself fixing just the symptoms of a problem, only for the situation to get much worse?

To be an effective problem-solver, you need to be systematic and logical in your approach. This quiz helps you assess your current approach to problem solving. By improving this, you'll make better overall decisions. And as you increase your confidence with solving problems, you'll be less likely to rush to the first solution – which may not necessarily be the best one.

Once you've completed the quiz, we'll direct you to tools and resources that can help you make the most of your problem-solving skills.

How Good Are You at Solving Problems?

Instructions.

For each statement, click the button in the column that best describes you. Please answer questions as you actually are (rather than how you think you should be), and don't worry if some questions seem to score in the 'wrong direction'. When you are finished, please click the 'Calculate My Total' button at the bottom of the test.

Answering these questions should have helped you recognize the key steps associated with effective problem solving.

This quiz is based on Dr Min Basadur's Simplexity Thinking problem-solving model. This eight-step process follows the circular pattern shown below, within which current problems are solved and new problems are identified on an ongoing basis. This assessment has not been validated and is intended for illustrative purposes only.

Below, we outline the tools and strategies you can use for each stage of the problem-solving process. Enjoy exploring these stages!

Step 1: Find the Problem (Questions 7, 12)

Some problems are very obvious, however others are not so easily identified. As part of an effective problem-solving process, you need to look actively for problems – even when things seem to be running fine. Proactive problem solving helps you avoid emergencies and allows you to be calm and in control when issues arise.

These techniques can help you do this:

PEST Analysis helps you pick up changes to your environment that you should be paying attention to. Make sure too that you're watching changes in customer needs and market dynamics, and that you're monitoring trends that are relevant to your industry.

Risk Analysis helps you identify significant business risks.

Failure Modes and Effects Analysis helps you identify possible points of failure in your business process, so that you can fix these before problems arise.

After Action Reviews help you scan recent performance to identify things that can be done better in the future.

Where you have several problems to solve, our articles on Prioritization and Pareto Analysis help you think about which ones you should focus on first.

Step 2: Find the Facts (Questions 10, 14)

After identifying a potential problem, you need information. What factors contribute to the problem? Who is involved with it? What solutions have been tried before? What do others think about the problem?

If you move forward to find a solution too quickly, you risk relying on imperfect information that's based on assumptions and limited perspectives, so make sure that you research the problem thoroughly.

Step 3: Define the Problem (Questions 3, 9)

Now that you understand the problem, define it clearly and completely. Writing a clear problem definition forces you to establish specific boundaries for the problem. This keeps the scope from growing too large, and it helps you stay focused on the main issues.

A great tool to use at this stage is CATWOE . With this process, you analyze potential problems by looking at them from six perspectives, those of its Customers; Actors (people within the organization); the Transformation, or business process; the World-view, or top-down view of what's going on; the Owner; and the wider organizational Environment. By looking at a situation from these perspectives, you can open your mind and come to a much sharper and more comprehensive definition of the problem.

Cause and Effect Analysis is another good tool to use here, as it helps you think about the many different factors that can contribute to a problem. This helps you separate the symptoms of a problem from its fundamental causes.

Step 4: Find Ideas (Questions 4, 13)

With a clear problem definition, start generating ideas for a solution. The key here is to be flexible in the way you approach a problem. You want to be able to see it from as many perspectives as possible. Looking for patterns or common elements in different parts of the problem can sometimes help. You can also use metaphors and analogies to help analyze the problem, discover similarities to other issues, and think of solutions based on those similarities.

Traditional brainstorming and reverse brainstorming are very useful here. By taking the time to generate a range of creative solutions to the problem, you'll significantly increase the likelihood that you'll find the best possible solution, not just a semi-adequate one. Where appropriate, involve people with different viewpoints to expand the volume of ideas generated.

Tip: Don't evaluate your ideas until step 5. If you do, this will limit your creativity at too early a stage.

Step 5: Select and Evaluate (Questions 6, 15)

After finding ideas, you'll have many options that must be evaluated. It's tempting at this stage to charge in and start discarding ideas immediately. However, if you do this without first determining the criteria for a good solution, you risk rejecting an alternative that has real potential.

Decide what elements are needed for a realistic and practical solution, and think about the criteria you'll use to choose between potential solutions.

Paired Comparison Analysis , Decision Matrix Analysis and Risk Analysis are useful techniques here, as are many of the specialist resources available within our Decision-Making section . Enjoy exploring these!

Step 6: Plan (Questions 1, 16)

You might think that choosing a solution is the end of a problem-solving process. In fact, it's simply the start of the next phase in problem solving: implementation. This involves lots of planning and preparation. If you haven't already developed a full Risk Analysis in the evaluation phase, do so now. It's important to know what to be prepared for as you begin to roll out your proposed solution.

The type of planning that you need to do depends on the size of the implementation project that you need to set up. For small projects, all you'll often need are Action Plans that outline who will do what, when, and how. Larger projects need more sophisticated approaches – you'll find out more about these in the article What is Project Management? And for projects that affect many other people, you'll need to think about Change Management as well.

Here, it can be useful to conduct an Impact Analysis to help you identify potential resistance as well as alert you to problems you may not have anticipated. Force Field Analysis will also help you uncover the various pressures for and against your proposed solution. Once you've done the detailed planning, it can also be useful at this stage to make a final Go/No-Go Decision , making sure that it's actually worth going ahead with the selected option.

Step 7: Sell the Idea (Questions 5, 8)

As part of the planning process, you must convince other stakeholders that your solution is the best one. You'll likely meet with resistance, so before you try to “sell” your idea, make sure you've considered all the consequences.

As you begin communicating your plan, listen to what people say, and make changes as necessary. The better the overall solution meets everyone's needs, the greater its positive impact will be! For more tips on selling your idea, read our article on Creating a Value Proposition and use our Sell Your Idea Skillbook.

Step 8: Act (Questions 2, 11)

Finally, once you've convinced your key stakeholders that your proposed solution is worth running with, you can move on to the implementation stage. This is the exciting and rewarding part of problem solving, which makes the whole process seem worthwhile.

This action stage is an end, but it's also a beginning: once you've completed your implementation, it's time to move into the next cycle of problem solving by returning to the scanning stage. By doing this, you'll continue improving your organization as you move into the future.

Problem solving is an exceptionally important workplace skill.

Being a competent and confident problem solver will create many opportunities for you. By using a well-developed model like Simplexity Thinking for solving problems, you can approach the process systematically, and be comfortable that the decisions you make are solid.

Given the unpredictable nature of problems, it's very reassuring to know that, by following a structured plan, you've done everything you can to resolve the problem to the best of your ability.

This assessment has not been validated and is intended for illustrative purposes only. It is just one of many Mind Tool quizzes that can help you to evaluate your abilities in a wide range of important career skills.

If you want to reproduce this quiz, you can purchase downloadable copies in our Store .

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Problem Solving Skills: Performance Review Examples (Rating 1 – 5)

By Status.net Editorial Team on July 21, 2023 — 4 minutes to read

Problem solving is an important skill in any work environment: it includes the ability to identify, understand, and develop solutions to complex issues while maintaining a focus on the end goal. Evaluating this skill in employees during performance reviews can be highly beneficial for both the employee and the organization.

Questions that can help you determine an employee’s rating for problem solving skills:

• How well does the employee define the problem and identify its root cause?
• How creative is the employee in generating potential solutions?
• How effective is the employee in implementing the chosen solution?
• How well does the employee evaluate the effectiveness of the solution and adjust it if necessary?

Related: Best Performance Review Examples for 48 Key Skills

2000+ Performance Review Phrases: The Complete List (Performance Feedback Examples)

Performance Review Phrases and Paragraphs Examples For Problem Solving

5 – outstanding.

Phrases examples:

• Consistently demonstrates exceptional problem-solving abilities
• Proactively identifies issues and offers innovative solutions
• Quickly adapts to unforeseen challenges and finds effective resolutions
• Exceptional problem-solving ability, consistently providing innovative solutions
• Regularly goes above and beyond to find creative solutions to complicated issues
• Demonstrates a keen understanding of complex problems and quickly identifies effective solutions

Paragraph Example 1

“Jane consistently demonstrates outstanding problem-solving skills. She proactively identifies issues in our department and offers innovative solutions that have improved processes and productivity. Her ability to quickly adapt to unforeseen challenges and find effective resolutions is commendable and has proven invaluable to the team.”

Paragraph Example 2

“Sarah has demonstrated an outstanding ability in problem solving throughout the year. Her innovative solutions have significantly improved our department’s efficiency, and she consistently goes above and beyond expectations to find creative approaches to complicated issues.”

4 – Exceeds Expectations

• Demonstrates a strong aptitude for solving complex problems
• Often takes initiative in identifying and resolving issues
• Effectively considers multiple perspectives and approaches before making decisions
• Displayed a consistently strong ability to tackle challenging problems efficiently
• Often takes the initiative to solve problems before they escalate
• Demonstrates a high level of critical thinking when resolving issues

“John exceeds expectations in problem-solving. He has a strong aptitude for solving complex problems and often takes initiative in identifying and resolving issues. His ability to consider multiple perspectives and approaches before making decisions has led to valuable improvements within the team.”

“Sam consistently exceeded expectations in problem solving this year. His efficient handling of challenging issues has made a positive impact on our team, and he often takes the initiative to resolve problems before they escalate. Sam’s critical thinking ability has been a valuable asset to our organization, and we appreciate his efforts.”

3 – Meets Expectations

• Displays adequate problem-solving skills when faced with challenges
• Generally able to identify issues and propose viable solutions
• Seeks assistance when necessary to resolve difficult situations
• Demonstrates a solid understanding of problem-solving techniques
• Capable of resolving everyday issues independently
• Shows perseverance when facing difficult challenges

“Mary meets expectations in her problem-solving abilities. She displays adequate skills when faced with challenges and is generally able to identify issues and propose viable solutions. Mary also seeks assistance when necessary to resolve difficult situations, demonstrating her willingness to collaborate and learn.”

“Sarah meets expectations in her problem-solving abilities. She demonstrates a solid understanding of problem-solving techniques and can resolve everyday issues independently. We value her perseverance when facing difficult challenges and encourage her to continue developing these skills.”

2 – Needs Improvement

• Struggles to find effective solutions to problems
• Tends to overlook critical details when evaluating situations
• Reluctant to seek help or collaborate with others to resolve issues
• Struggles to find effective solutions when faced with complex issues
• Often relies on assistance from others to resolve problems
• May lack confidence in decision-making when solving problems

“Tom’s problem-solving skills need improvement. He struggles to find effective solutions to problems and tends to overlook critical details when evaluating situations. Tom should work on being more willing to seek help and collaborate with others to resolve issues, which will ultimately strengthen his problem-solving abilities.”

“Mark’s problem-solving skills need improvement. He often struggles to find effective solutions for complex issues and seeks assistance from others to resolve problems. We encourage Mark to build his confidence in decision-making and focus on developing his problem-solving abilities.”

1 – Unacceptable

• Fails to identify and resolve problems in a timely manner
• Lacks critical thinking skills necessary for effective problem-solving
• Often creates additional issues when attempting to resolve problems
• Demonstrates a consistent inability to resolve even basic issues
• Often avoids responsibility for problem-solving tasks
• Fails to analyze problems effectively, leading to poor decision-making

“Sally’s problem-solving skills are unacceptable. She consistently fails to identify and resolve problems in a timely manner, and her lack of critical thinking skills hinders her ability to effectively solve challenges. Additionally, her attempts to resolve problems often create additional issues, resulting in a negative impact on the team’s overall performance.”

“Susan’s problem-solving performance has been unacceptable this year. She consistently demonstrates an inability to resolve basic issues and avoids taking responsibility for problem-solving tasks. Her ineffectiveness in analyzing problems has led to poor decision-making. It is crucial that Susan improve her problem-solving skills to succeed in her role.”

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Check Your Understanding

Answer: d = 1720 m

Answer: a = 8.10 m/s/s

Answers: d = 33.1 m and v f = 25.5 m/s

Answers: a = 11.2 m/s/s and d = 79.8 m

Answer: t = 1.29 s

Answers: a = 243 m/s/s

Answer: a = 0.712 m/s/s

Answer: d = 704 m

Answer: d = 28.6 m

Answer: v i = 7.17 m/s

Answer: v i = 5.03 m/s and hang time = 1.03 s (except for in sports commericals)

Answer: a = 1.62*10 5 m/s/s

Answer: d = 48.0 m

Answer: t = 8.69 s

Answer: a = -1.08*10^6 m/s/s

Answer: d = -57.0 m (57.0 meters deep) 

Answer: v i = 47.6 m/s

Answer: a = 2.86 m/s/s and t = 30. 8 s

Answer: a = 15.8 m/s/s

Answer: v i = 94.4 mi/hr

Solutions to Above Problems

d = (0 m/s)*(32.8 s)+ 0.5*(3.20 m/s 2 )*(32.8 s) 2

Return to Problem 1

110 m = (0 m/s)*(5.21 s)+ 0.5*(a)*(5.21 s) 2

110 m = (13.57 s 2 )*a

a = (110 m)/(13.57 s 2 )

a = 8.10 m/ s 2

Return to Problem 2

d = (0 m/s)*(2.60 s)+ 0.5*(-9.8 m/s 2 )*(2.60 s) 2

d = -33.1 m (- indicates direction)

v f = v i + a*t

v f = 0 + (-9.8 m/s 2 )*(2.60 s)

v f = -25.5 m/s (- indicates direction)

Return to Problem 3

a = (46.1 m/s - 18.5 m/s)/(2.47 s)

a = 11.2 m/s 2

d = v i *t + 0.5*a*t 2

d = (18.5 m/s)*(2.47 s)+ 0.5*(11.2 m/s 2 )*(2.47 s) 2

d = 45.7 m + 34.1 m

(Note: the d can also be calculated using the equation v f 2 = v i 2 + 2*a*d)

Return to Problem 4

-1.40 m = (0 m/s)*(t)+ 0.5*(-1.67 m/s 2 )*(t) 2

-1.40 m = 0+ (-0.835 m/s 2 )*(t) 2

(-1.40 m)/(-0.835 m/s 2 ) = t 2

1.68 s 2 = t 2

Return to Problem 5

a = (444 m/s - 0 m/s)/(1.83 s)

a = 243 m/s 2

d = (0 m/s)*(1.83 s)+ 0.5*(243 m/s 2 )*(1.83 s) 2

d = 0 m + 406 m

Return to Problem 6

(7.10 m/s) 2 = (0 m/s) 2 + 2*(a)*(35.4 m)

50.4 m 2 /s 2 = (0 m/s) 2 + (70.8 m)*a

(50.4 m 2 /s 2 )/(70.8 m) = a

a = 0.712 m/s 2

Return to Problem 7

(65 m/s) 2 = (0 m/s) 2 + 2*(3 m/s 2 )*d

4225 m 2 /s 2 = (0 m/s) 2 + (6 m/s 2 )*d

(4225 m 2 /s 2 )/(6 m/s 2 ) = d

Return to Problem 8

d = (22.4 m/s + 0 m/s)/2 *2.55 s

d = (11.2 m/s)*2.55 s

Return to Problem 9

(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(2.62 m)

0 m 2 /s 2 = v i 2 - 51.35 m 2 /s 2

51.35 m 2 /s 2 = v i 2

v i = 7.17 m/s

Return to Problem 10

(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(1.29 m)

0 m 2 /s 2 = v i 2 - 25.28 m 2 /s 2

25.28 m 2 /s 2 = v i 2

v i = 5.03 m/s

To find hang time, find the time to the peak and then double it.

0 m/s = 5.03 m/s + (-9.8 m/s 2 )*t up

-5.03 m/s = (-9.8 m/s 2 )*t up

(-5.03 m/s)/(-9.8 m/s 2 ) = t up

t up = 0.513 s

hang time = 1.03 s

Return to Problem 11

(521 m/s) 2 = (0 m/s) 2 + 2*(a)*(0.840 m)

271441 m 2 /s 2 = (0 m/s) 2 + (1.68 m)*a

(271441 m 2 /s 2 )/(1.68 m) = a

a = 1.62*10 5 m /s 2

Return to Problem 12

• (NOTE: the time required to move to the peak of the trajectory is one-half the total hang time - 3.125 s.)

First use:  v f  = v i  + a*t

0 m/s = v i  + (-9.8  m/s 2 )*(3.13 s)

0 m/s = v i  - 30.7 m/s

v i  = 30.7 m/s  (30.674 m/s)

Now use:  v f 2  = v i 2  + 2*a*d

(0 m/s) 2  = (30.7 m/s) 2  + 2*(-9.8  m/s 2 )*(d)

0 m 2 /s 2  = (940 m 2 /s 2 ) + (-19.6  m/s 2 )*d

-940  m 2 /s 2  = (-19.6  m/s 2 )*d

(-940  m 2 /s 2 )/(-19.6  m/s 2 ) = d

Return to Problem 13

-370 m = (0 m/s)*(t)+ 0.5*(-9.8 m/s 2 )*(t) 2

-370 m = 0+ (-4.9 m/s 2 )*(t) 2

(-370 m)/(-4.9 m/s 2 ) = t 2

75.5 s 2 = t 2

Return to Problem 14

(0 m/s) 2 = (367 m/s) 2 + 2*(a)*(0.0621 m)

0 m 2 /s 2 = (134689 m 2 /s 2 ) + (0.1242 m)*a

-134689 m 2 /s 2 = (0.1242 m)*a

(-134689 m 2 /s 2 )/(0.1242 m) = a

a = -1.08*10 6 m /s 2

(The - sign indicates that the bullet slowed down.)

Return to Problem 15

d = (0 m/s)*(3.41 s)+ 0.5*(-9.8 m/s 2 )*(3.41 s) 2

d = 0 m+ 0.5*(-9.8 m/s 2 )*(11.63 s 2 )

d = -57.0 m

(NOTE: the - sign indicates direction)

Return to Problem 16

(0 m/s) 2 = v i 2 + 2*(- 3.90 m/s 2 )*(290 m)

0 m 2 /s 2 = v i 2 - 2262 m 2 /s 2

2262 m 2 /s 2 = v i 2

v i = 47.6 m /s

Return to Problem 17

( 88.3 m/s) 2 = (0 m/s) 2 + 2*(a)*(1365 m)

7797 m 2 /s 2 = (0 m 2 /s 2 ) + (2730 m)*a

7797 m 2 /s 2 = (2730 m)*a

(7797 m 2 /s 2 )/(2730 m) = a

a = 2.86 m/s 2

88.3 m/s = 0 m/s + (2.86 m/s 2 )*t

(88.3 m/s)/(2.86 m/s 2 ) = t

t = 30. 8 s

Return to Problem 18

( 112 m/s) 2 = (0 m/s) 2 + 2*(a)*(398 m)

12544 m 2 /s 2 = 0 m 2 /s 2 + (796 m)*a

12544 m 2 /s 2 = (796 m)*a

(12544 m 2 /s 2 )/(796 m) = a

a = 15.8 m/s 2

Return to Problem 19

v f 2 = v i 2 + 2*a*d

(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(91.5 m)

0 m 2 /s 2 = v i 2 - 1793 m 2 /s 2

1793 m 2 /s 2 = v i 2

v i = 42.3 m/s

Now convert from m/s to mi/hr:

v i = 42.3 m/s * (2.23 mi/hr)/(1 m/s)

v i = 94.4 mi/hr

Return to Problem 20

14 Inspirational Examples of Business Ideas That Solve Problems

By Brett Farmiloe

Problems are the spark for innovative solutions. Here we've gathered the accounts of entrepreneurs who transformed challenges into successful business ventures. Read on to learn why these founders believe in the power of starting businesses to solve common problems.

14 businesses that were founded to solve a problem

1. the adu guide.

Startup story: "My journey began when I came across the widespread issue of limited housing options. Recognizing the need for adaptable living spaces, I established a construction company that specializes in accessory dwelling units (ADUs). I understand the importance of addressing common problems with innovative solutions. The ADU Guide not only offers homeowners expert advice from licensed consultants, but also helps to alleviate the housing crisis."

Takeaway: "Starting a business to address common issues not only promotes personal success, but also benefits communities by providing practical and meaningful solutions."

— Eli Cohen , The ADU Guide

2. Runtofly

Startup story: " After finishing my studies a few years back, I decided to reward myself with a trip somewhere. I didn’t know exactly where to go, and I was ready to fly anywhere (well…almost!), only if the price was really good. I was literally ready to depart 'now'! I thought it was exactly the idea of a 'last-minute flight' that many people talk about. But unfortunately (or actually, fortunately!), I found out that most flights departing in the next hours or days were very expensive, even when many seats were still available. I realized that cheap last-minute flights were only a myth and that there was a problem of untapped supply and unaddressed demand.

"Instead of accepting things as they were, I made up my mind to work on this problem and make last-minute flights real. It took a few years of blood, sweat, and tears, but finally, I was able to see my vision come to reality in 2024 when Runtofly.co.uk went live."

Takeaway: "People should always start businesses to solve a problem that others have. And the most successful businesses are the ones where the founder experienced the problem themselves firsthand. Too many entrepreneurs start a company just for the sake of starting a company without starting from a problem. Instead, they end up with a 'solution in search of a problem,' instead of the other way around."

— Federico Grimaccia , Runtofly

3. True Friends Moving Company

Startup story: " My entrepreneurial journey began with a rather simple observation—while living in a close-knit community, I noticed a recurring issue many faced: the lack of professional moving services. Can you imagine the stress of moving without reliable help? The distress calls for movers were on the rise, and although several informal services existed, none could instill the confidence and care one would expect during such a stressful time.

"I started small—just me and one truck. But, as we helped families move homes, our reputation grew. Fast-forward to today, and we're not just a moving company; we've become a trusted partner in new beginnings. Solving a common problem isn't just about spotting it; it's about being passionate about providing a solution that people will trust and remember.

Takeaway: "Why should others start businesses to solve common problems? Here's a compelling reason: genuine market demand. When you alleviate a pain point that many face, you immediately tap into an ecosystem where every satisfied client can become a vocal advocate. Growth rooted in solving real-world problems tends to be organic, sustained, and deeply rewarding.

"In my case, as I performed each move, I refined my service and built relationships that turned customers into effective brand ambassadors. My business grew beyond a single truck through the most powerful marketing tool—word-of-mouth referrals.

"When you solve a common problem, you do more than start a business; you build a community cornerstone that supports and grows with its people. Entrepreneurs should be encouraged to look into their own communities and identify needs because, more often than not, serving your community leads to real impact and sustainable success."

— Chris Knowles , True Friends Moving Company

4. InBound Blogging

Startup story: "Initially, when I started my business, I wanted to provide go-to solutions for people who wanted to monetize their blogs. Many small creators and businesses struggle to build websites that generate an income. Recognizing this gap, I started InBound Blogging to provide tailored solutions and help bloggers make a living from their content. My agency has since grown, has grown, and we've pivoted to offering comprehensive SEO services, specifically for B2B SaaS companies."

Takeaway: "Building a business around the concept of problem-solving works because, essentially, you've already got a target audience. If there is a market need for the solutions you're providing, people will seek you out for your expertise to help them out. What you have to do is create something of value so you can build a successful and sustainable company. It gives you space to carve out a niche of your own, and over time, you can do what I did and branch out and offer services in other areas."

— Nikola Baldikov , InBound Blogging

5. SellCoursesOnline

Startup story: "During my years in the digital marketing agency space, I saw firsthand many business owners struggling with the tech side of things. Sure, there's the internet, but it's also filled with a lot of fluffy advice, contradictory information, and even misinformation. It became apparent to me that navigating the tech side of growing an online business presence isn't an easy and straightforward task for everyone. It's a problem in the market that needed a solution, and so I provided one.

"This is how our agency was born. It is essentially a one-stop resource for entrepreneurs and creators to find the support, technical guidance, and the right tools they need to build their e-learning platforms and online courses."

Takeaway: "Simple and common problems like this often get overlooked, but there are actually big opportunities for observant business people who are smart enough to provide a solution."

— Baidhurya Mani , SellCoursesOnline

6. Pool Care Arizona LLC

Startup story: "My foray into entrepreneurship commenced when I encountered obstacles in locating dependable and effective pool maintenance services for my personal residence. Motivated by the dearth of alternatives that satisfied my criteria, I perceived this as a chance to address a market void and offer superior pool services to individuals encountering comparable predicaments."

Takeaway: "Entrepreneurs can address unfulfilled demands, generate value for customers, and make a positive impact on their communities by establishing companies that resolve prevalent issues. Entrepreneurs can create enterprises that not only prosper but also have a significant social and economic impact by recognizing areas of dissatisfaction and presenting novel remedies. This collective experience highlights the criticality of entrepreneurship as a catalyst for advancement and a solution to societal issues."

— Jacob Mendrin , Pool Care Arizona LLC

7. humble help

Startup story: "Starting a business isn't just about making money—it's about finding solutions to real problems. My journey began during the tough times of 2020 when I saw so many small businesses struggling to survive. That's when Humble Help Studio came to life, with a mission to offer support and relief to these important local players."

Takeaway: "Being an entrepreneur means looking at problems as chances to make something better. I was inspired by the hardships these businesses were going through, and it made me realize how important it is to care deeply about the issues you're trying to solve. The best reason to start a business is wanting to make a difference, especially for problems that affect the people and places you love. This mindset turns obstacles into opportunities for making things better, sparking innovation and having a positive impact.

"Simply put, entrepreneurs have this special ability to fix what's broken, using their ideas, hard work, and passion. This doesn't just lead to successful businesses; it helps build a stronger, kinder community."

— Vick Antonyan , humble help

Startup story: "I started Chadix because I was fed up with how complex and time-consuming most SEO tools were. As an online marketer trying to grow sites for myself and clients, I wasted tons of hours on manual content creation, optimizations, keyword research, etc. The lightbulb moment was realizing these SEO processes could be automated using AI, freeing up my time for strategy and scaling.

"So, I built a company to simplify and optimize SEO—with AI automatically handling content production, insights, optimizations, and more. I didn't have a waitlist of thousands at first. Leveraging my personal brand on Facebook, I created the Chadix Facebook Group to build an audience interested in my idea. Within months, I organically grew the group to 850 members.

"To validate Chadix's potential, I invited engaged group members to become Alpha testers. Nearly 80 people signed up and provided invaluable feedback on the software prototyping and UI. The tiny churn rate showed I struck a nerve by addressing the universal pain points around SEO complexity that most marketers and businesses face."

Takeaway: "I believe solving real problems people encounter is the best way to build an impactful business. If your solution alleviates frustrations, saves money, or saves time for customers, you inherently create tremendous value. Chadix was born out of my personal annoyance with SEO tools—then validated by other marketers feeling the same pain. Addressing universal problems with creative solutions is every entrepreneur's path to success."

— Danny Veiga , Chadix

9. My Millennial Guide

Startup story: "When I graduated from college, I was excited to start my career and life as an independent adult. However, I soon found myself weighed down by over $30,000 in student loan debt. No matter how hard I worked at my finance job, I felt like I was drowning in interest payments and getting nowhere.

"I realized so many other millennials were trapped in this same cycle of debt. I knew there had to be a way to break free. So, I started educating myself on personal finance, cutting unnecessary expenses, developing side hustles, and putting every extra penny toward paying down the principal. Through determination and sacrifice, I managed to be completely debt-free in just one year.

"That experience showed me how empowering it is to solve a critical personal problem through resourcefulness and grit. I started My Millennial Guide to help other young people trapped by debt or financial struggles. By sharing what I learned, I hoped to give them the knowledge and inspiration to take control of their own situation."

Takeaway: "Starting a business to address a common problem you've personally faced allows you to intimately understand customer needs. Your own journey also gives you credibility and passion to persevere. I think if you see an unmet need out there, you have both an opportunity and a responsibility to find a solution. Turn your big idea into tangible help for those still struggling. Allow your purpose-driven business to improve lives while also achieving your dreams."

— Brian Meiggs , My Millennial Guide

10. Perfect Locks

Startup story: "During my childhood in India, I faced a lot of bullying and a lot of pressure from society about hair and beauty norms. This personal experience gave me the idea for Perfect Locks. I'd seen firsthand how these issues can impact people, and I wanted to create a company that provides hair extensions and gives people the tools they need to look and feel their best."

Takeaway: "My business was started out of a passion to solve a common issue, creating solutions that boost self-confidence and empower people to express themselves. Running a business to solve everyday issues is important because it enables entrepreneurs to make a positive difference in the world while meeting a need in the marketplace.

"By recognizing and solving problems that most people face, entrepreneurs can develop innovative solutions that make a difference in people's lives and in the world. In addition, running a business focused on solving common issues can be extremely rewarding both personally and professionally. It allows you to combine enthusiasm with purpose, motivating you to overcome challenges and reach your goals.

"Entrepreneurship allows people to do good, build value, and build a legacy by solving common problems and issues facing society. It’s not just about making money; it’s about making a difference and leaving a better world."

— Priyanka Swamy , Perfect Locks

11. Life Architekture

Startup story: "As a life coach, my journey began when I noticed a common problem: many people feel lost, without direction or clarity about their personal and professional lives. Motivated by a desire to help others find their path, I started my life-coaching business. My goal is to provide guidance, support, and strategies to help my clients uncover their passions and find meaning in their lives."

Takeaway: "I believe that starting a business is more than just filling a market gap; it's about making a real difference in people's lives. For me, the decision to become a life coach was driven by the impact I knew I could make. As entrepreneurs, we have the opportunity to transform personal insights and experiences into solutions that can benefit others. By addressing common challenges, we create positive change and empower people to have the support they need to thrive and succeed."

— Bayu Prihandito , Life Architekture

12. Yarooms

Startup story: "Yarooms got started nearly a dozen years ago when I got so sick of not knowing whether a conference room was open at my corporate programming job and decided to stick a tablet to a wall to let people mark it on a calendar. The need to solve this one annoying problem has allowed me to see other small issues that make working in the office, at home, or in a hybrid setting less efficient and enjoyable, and has led me to understand the small fixes we can do to develop solutions."

Takeaway: "It sounds silly, but if you can't find an available solution to your problem or annoyance, fix it yourself. I guarantee if it bothers you, it bothers many other people too."

— Dragos Badea , Yarooms

13. Davis Business Law

Startup story: "I started Davis Business Law to solve three problems: how law firms treat clients, how attorneys get treated in firms, and how attorneys treat the firm’s staff.

"The two things clients really hate about law firms are getting the 'mushroom treatment,' which is getting fed BS while being kept in the dark, and 'blank check syndrome,' where the client feels like they are handing over a blank check for the firm to choose the amount. We deal with these issues through a disciplined case finance where we let our clients know what work is coming down the pike and have them prepay for it if they want us to proceed. Alternatively, it gives them a chance to talk to their attorney and then decide. It takes a lot of work, but counterintuitively, it has led to a massive increase in client satisfaction.

"Law firms also tend to treat associates very poorly with excessive billing demands. Nine hours per day is not out of the ordinary. This is largely a symptom of bad financial management. We run very efficiently and routinely match the salaries of these firms at our six-hour-per-day expectation. This leads to happy lawyers with work-life balance. The kicker is they turn out better legal work because they are not stressed.

"The staff at a lot of law firms suffer from professional arrogance and disrespect from the lawyers. This is not ubiquitous, but it is common. Our hiring process focuses on weeding out attorneys with these attitudes and reinforcing our culture of respect."

Takeaway: "Dealing head-on with these three problems is the foundation of our rapid growth. There is an old saying, 'If you build a better mousetrap, the world will beat a path to your door.' That is the essence of starting a new business. You find a problem and create a solution. In our line of work, we were lucky enough to have three problems and diligent enough to develop three fixes. Implementing them is our recipe for success."

— Matthew Davis , Davis Business Law

Startup story: "As a consultant in the industrial sector, I often worked with businesses looking to diversify. The trouble was that women, LGBTQ, and BIPOC workers had been marginalized so deeply, they weren't even on the radar of most recruiting firms. It was an obvious gap in the market—I knew qualified but neglected workers were out there; they just needed to connect with the market. So, I launched Bemana, a female-majority recruiting firm specializing in the industrial and equipment sector. It wasn't about politics; it was about seeing a gap in the profession I wanted to rectify."

Takeaway: "The response was more than I expected. Not only did a variety of candidates feel comfortable working with me, but clients also began to think of me as someone progressing the sector. That forward-thinking identity helped me stand out from the crowd."

— Linn Atiyeh , Bemana

About the Author

Post by: Brett Farmiloe

Brett Farmiloe is the founder and CEO of Featured, a platform where business leaders can answer questions related to their expertise and get published in articles featuring their insights.

Company: Featured

Website: www.featured.com

Connect with me on LinkedIn .

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8D Corrective Action: Mastering Problem-Solving for Continuous Improvement

May 13th, 2024

Businesses constantly refine products, services, and workflows to stay ahead. But issues can still pop up, angering customers and jacking costs while hurting a company’s image. This is where the 8D corrective action problem-solving method earns its stripes.

It was developed by Ford in the 80s and has since spread widely across manufacturing, healthcare, aerospace, and more.

The 8D approach is a methodical process combining pros from different parts of the company, analytical tools, and fact-based decision-making.

By following its eight systematic steps, organizations can expertly handle thorny problems. They uncover root causes and implement lasting fixes addressing immediate concerns while fueling constant upgrades to prevent repeat issues.

Key Highlights

• Understanding the origins and history of the 8D corrective action methodology, its benefits, and when to apply it for optimal results.
• Exploring the eight disciplined steps of the 8D corrective action process.
• Integrating the 8D methodology with quality management systems, leveraging Enterprise Quality Management Software (EQMS) to streamline workflows.
• Examining case studies and examples from various industries, including manufacturing, service, healthcare, and the automotive sector.

Understanding the 8D Corrective Action Problem-Solving Methodology

The Eight Disciplines (8D) methodology is a structured, team-based approach to problem-solving that aims to identify the root causes of issues and implement effective corrective actions. 

It is a comprehensive framework that combines analytical tools, cross-functional collaboration, and a disciplined mindset to tackle complex problems systematically.

The 8D process establishes a step-by-step approach that guides organizations through eight distinct disciplines, each building upon the previous one. 

Origins and History of 8D Corrective Action

The origins of the 8D methodology can be traced back to the 1980s when it was developed and pioneered by Ford Motor Company. 

Initially referred to as “ Team Oriented Problem Solving ” (TOPS), this approach was designed to address the recurring quality issues that plagued the automotive industry at the time.

Recognizing the limitations of traditional problem-solving techniques, Ford sought to establish a more robust and effective framework that would not only resolve immediate concerns but also drive continuous improvement and prevent future issues. 

The 8D methodology quickly gained traction within Ford and was subsequently adopted as the company’s primary approach for documenting and addressing problem-solving efforts.

As the benefits of the 8D corrective action process became evident, it rapidly gained popularity among other manufacturers and industries, transcending its automotive roots. 

Today, the 8D methodology is widely employed across various sectors, including manufacturing, healthcare, aerospace, and service industries, among others.

Benefits of Using 8D Corrective Action

Implementing the 8D problem-solving methodology offers numerous benefits to organizations, including:

1. Systematic Approach : The structured nature of the 8D process ensures a consistent and comprehensive approach to problem-solving, reducing the risk of overlooking critical factors or jumping to premature conclusions.

2. Root Cause Identification : By emphasizing root cause analysis , the 8D methodology goes beyond addressing surface-level symptoms and focuses on identifying and eliminating the underlying causes of problems.

3. Cross-Functional Collaboration : The team-based approach fosters cross-functional collaboration, leveraging diverse perspectives and expertise from various departments, leading to more robust and well-rounded solutions.

4. Preventive Measures : The 8D corrective action process incorporates preventive actions to mitigate the recurrence of similar issues, promoting a culture of continuous improvement and proactive problem-solving.

5. Improved Quality and Reliability : By addressing root causes and implementing corrective actions, organizations can enhance the quality and reliability of their products, services, and processes, leading to increased customer satisfaction and cost savings.

6. Knowledge Sharing and Organizational Learning : The documentation and archiving of 8D processes facilitate knowledge sharing and organizational learning, enabling teams to build upon past experiences and lessons learned.

When to Apply 8D Corrective Action

The 8D problem-solving methodology is particularly valuable in situations where:

• Root Cause Analysis is Required: When issues persist despite initial troubleshooting efforts, or when the underlying causes are not immediately apparent, the 8D process can provide a structured approach to root cause analysis.
• Recurring Problems: If an organization experiences recurring problems or quality issues, the 8D methodology can help identify and eliminate the root causes, preventing future occurrences.
• Quality Issues with Significant Impact: When quality issues have a substantial impact on customer satisfaction, safety, regulatory compliance, or financial performance, the rigorous 8D approach can be employed to address the problem comprehensively.
• Complex Problems: For intricate problems involving multiple factors, processes, or departments, the cross-functional nature of the 8D team and the systematic approach can facilitate a thorough investigation and effective solution development.

By understanding the core principles, benefits, and appropriate application scenarios of the 8D problem-solving methodology, organizations can leverage this powerful framework to drive continuous improvement , enhance quality, and maintain a competitive edge in their respective industries.

The Eight Disciplines (8D) Process

At the heart of the 8D corrective action methodology lies a structured, step-by-step approach that guides organizations through eight distinct disciplines. 

Each discipline builds upon the previous one, ensuring a thorough investigation, analysis, and resolution of the problem at hand.

The eight disciplines of the 8D process are designed to facilitate a systematic and disciplined approach to problem-solving, leveraging cross-functional collaboration, analytical tools, and data-driven decision-making. 

D0: Planning and Preparation

Before embarking on the 8D corrective action journey, proper planning and preparation are crucial. This initial step, often referred to as Discipline Zero (D0), lays the foundation for a successful problem-solving effort.

During D0, the team gathers relevant information about the problem, assesses the need for interim containment actions, and establishes the prerequisites for forming an effective cross-functional team. 

This stage involves collecting data on symptoms, identifying potential risks, and ensuring that the necessary resources and support are in place to execute the 8D process effectively.

D1: Team Formation

The first formal discipline of the 8D process focuses on assembling a cross-functional team with the collective knowledge, skills, and expertise required to tackle the problem at hand. 

Effective team formation is critical to the success of the 8D corrective action effort, as it ensures diverse perspectives and a comprehensive understanding of the issue.

During D1, team members are carefully selected from various departments or functions, such as product engineering, process engineering, quality assurance, and data analysis.

Best practices in team formation involve considering factors such as technical expertise, problem-solving skills, interpersonal abilities, and the availability and commitment of potential team members. 

Establishing ground rules, communication protocols, and team-building exercises can further enhance collaboration and effective teamwork.

D2: Problem Description

In Discipline 2, the team focuses on accurately describing the problem, utilizing quantitative data and evidence-based approaches. 

This step is crucial, as it establishes a shared understanding of the issue and guides the subsequent steps of the 8D process.

The problem description involves defining the problem statement in specific, measurable terms, identifying the affected product or process, and quantifying the impact on operations, quality, customer satisfaction, and costs. 

Tools such as the “ 5 Whys ” technique, Ishikawa (fishbone) diagrams , and “ Is/Is Not ” analysis can aid in this process, helping to capture relevant details and categorize information.

D3: Interim Containment Actions

While the team works towards identifying and implementing permanent solutions, Discipline 3 focuses on implementing interim containment actions to mitigate the immediate impact of the problem and protect customers from further exposure.

Interim containment actions are temporary measures designed to isolate the problem and prevent it from causing further harm or spreading to other areas, processes, or products. 

These actions may include segregating defective products, implementing additional inspections or checks, or introducing manual oversight until permanent corrective actions are in place.

It is essential to verify the effectiveness of interim containment actions and monitor their implementation to ensure that they are successful in containing the problem and minimizing its impact on operations and customers.

D4: Root Cause Analysis

At the core of the 8D corrective action process lies Discipline 4, which focuses on identifying the root causes of the problem through rigorous analysis and data-driven investigation. 

This step is crucial, as it lays the foundation for developing effective and sustainable corrective actions.

During root cause analysis, the team employs various analytical tools and techniques, such as comparative analysis , fault tree analysis , and root cause verification experiments. 

These methods help to isolate and verify the underlying causes of the problem, separating symptoms from true root causes.

Thorough documentation and verification of root causes are essential in this discipline, ensuring that the team has a solid foundation for developing effective corrective actions.

D5: Permanent Corrective Actions (PCAs)

Building upon the insights gained from root cause analysis , Discipline 5 focuses on selecting and verifying permanent corrective actions (PCAs) that address the identified root causes and mitigate the risk of future occurrences.

During this stage, the team evaluates potential corrective actions based on their effectiveness in addressing the root causes, as well as their feasibility, cost, and potential impact on other processes or systems. 

Risk assessment tools, such as Failure Mode and Effects Analysis (FMEA), can aid in this evaluation process.

Once the most appropriate corrective actions have been selected, the team verifies their effectiveness through pilot testing , simulations, or other validation methods. 

This step ensures that the proposed solutions will indeed resolve the problem and prevent its recurrence without introducing unintended consequences.

Detailed planning and documentation of the corrective actions, including acceptance criteria, implementation timelines, and responsibilities, are critical components of Discipline 5.

D6: Implementation and Validation

In Discipline 6, the team focuses on implementing the selected permanent corrective actions and validating their effectiveness in resolving the problem and preventing future occurrences.

This stage involves developing a comprehensive project plan that outlines the steps, timelines, and resources required for successful implementation. 

Effective communication and coordination with all relevant stakeholders, including cross-functional teams and management, are essential to ensure a smooth transition and minimize disruptions.

During implementation, the team closely monitors the progress and performance of the corrective actions, gathering data and feedback to validate their effectiveness. 

This validation process may involve conducting simulations, inspections, or collecting performance metrics to assess the impact of the implemented solutions.

If the validation process reveals any shortcomings or unintended consequences, the team may need to revisit the corrective actions, make adjustments, or conduct further root cause analysis to address any remaining issues.

D7: Preventive Actions

Discipline 7 of the 8D process focuses on taking preventive measures to ensure that the lessons learned and improvements made during the problem-solving journey are embedded into the organization’s processes, systems, and culture.

In this stage, the team reviews similar products, processes, or areas that could be affected by the same or similar root causes, identifying opportunities to apply preventive actions more broadly. 

This proactive approach helps to mitigate the risk of future occurrences and promotes a culture of continuous improvement .

Effective implementation of preventive actions requires cross-functional collaboration, clear communication, and ongoing monitoring to ensure their sustained effectiveness.

D8: Closure and Celebration

The final discipline of the 8D process, D8, serves as a critical step in recognizing the team’s efforts, sharing lessons learned, and celebrating the successful resolution of the problem.

During this stage, the team conducts a final review of the problem-solving journey, documenting key lessons and insights that can be applied to future projects. 

This documentation not only preserves institutional knowledge but also facilitates continuous improvement by enabling the organization to build upon past experiences.

Equally important is the recognition and celebration of the team’s achievements. By acknowledging the collective efforts, dedication, and collaboration of team members, organizations can foster a positive and supportive culture that values problem-solving and continuous improvement.

Formal recognition events, such as team presentations or awards ceremonies, can be organized to showcase the team’s accomplishments and highlight the impact of their work on the organization’s quality, customer satisfaction, and overall performance.

By completing the eight disciplines of the 8D process, organizations can effectively navigate complex problems, identify root causes, implement sustainable solutions, and establish a foundation for continuous improvement and organizational learning.

Integrating 8D Corrective Action with Quality Management Systems

While the 8D problem-solving methodology offers a robust framework for addressing quality issues and driving continuous improvement, its effectiveness can be further amplified by integrating it with an organization’s quality management systems . 

Leveraging enterprise-level software solutions can streamline the 8D process, enhance collaboration, and foster a culture of continuous improvement.

The Role of EQMS in 8D Corrective Action

Enterprise Quality Management Software (EQMS) plays a pivotal role in supporting the successful implementation of the 8D corrective action methodology. 

By utilizing an EQMS, teams can benefit from features such as:

• Standardized 8D Workflows: Pre-configured 8D workflows and templates ensure consistency and adherence to best practices, guiding teams through each discipline with clearly defined tasks, responsibilities, and timelines.
• Collaboration and Communication: EQMS platforms facilitate cross-functional collaboration by providing secure document sharing, real-time updates, and centralized communication channels, ensuring that all stakeholders remain informed and engaged throughout the 8D process.
• Data Management and Reporting: Comprehensive data management capabilities within an EQMS enable teams to easily capture, analyze, and report on quality data, facilitating data-driven decision-making and root cause analysis during the 8D process.
• Integration with Quality Systems: EQMS solutions often integrate with other quality management systems, such as corrective and preventive action (CAPA) systems, enabling seamless information sharing and ensuring that the insights gained from the 8D process are incorporated into broader quality improvement initiatives.

Automating 8D Corrective Action Workflows

One of the key advantages of leveraging an EQMS is the ability to automate 8D workflows, streamlining the process and reducing the administrative burden on teams. 

Automated workflows also facilitate consistent documentation and record-keeping, which is essential for maintaining compliance with industry regulations and standards, as well as enabling knowledge sharing and organizational learning.

Data-Driven Decision-making

The 8D corrective action methodology heavily relies on data-driven decision-making, particularly during the root cause analysis and corrective action selection phases. 

An EQMS provides teams with powerful data analysis and reporting capabilities, enabling them to quickly identify trends, patterns, and correlations that can inform their decision-making process.

Continuous Improvement Culture

Ultimately, the integration of the 8D methodology with an EQMS fosters a culture of continuous improvement within an organization. 

The insights gained from the 8D process, coupled with the robust reporting and analytics capabilities of an EQMS, provide organizations with a wealth of data and knowledge that can be leveraged to drive ongoing process optimization and quality enhancement initiatives.

Case Studies and Examples of 8D Corrective Action

To illustrate the practical application and impact of the 8D problem-solving methodology, let us explore a few real-world case studies and examples from various industries. 

These examples will showcase how organizations have successfully leveraged the 8D approach to address quality issues, resolve complex problems, and drive continuous improvement.

Manufacturing Quality Issues

In the manufacturing sector, where quality and reliability are paramount, the 8D methodology has proven invaluable in addressing a wide range of issues. 

One notable example is a leading automotive parts manufacturer that faced recurring quality issues with a critical component, resulting in costly rework and customer dissatisfaction.

By implementing the 8D process, a cross-functional team was assembled to investigate the problem. Through root cause analysis , they identified a flaw in the supplier’s raw material handling processes, leading to inconsistencies in the component’s material properties.

The team implemented interim containment actions to segregate and inspect incoming materials, while also working with the supplier to implement permanent corrective actions, such as upgrading their material handling equipment and revising their quality control procedures.

Service Industry Applications of 8D Corrective Action

While the 8D corrective action approach is often associated with manufacturing, it has also proven valuable in the service industry, where quality and process excellence are equally critical. 

A prominent financial institution faced challenges with excessive customer complaints related to billing errors and account discrepancies.

By implementing the 8D methodology, a cross-functional team analyzed the problem, identifying root causes such as outdated software systems, inadequate training for customer service representatives, and inefficient data entry processes.

The team implemented interim containment actions, including manual account audits and increased customer communication, while also developing permanent corrective actions, such as upgrading their billing software, revising training programs, and streamlining data entry procedures.

Healthcare and Life Sciences

In the healthcare and life sciences industries, where patient safety and regulatory compliance are paramount, the 8D methodology has proven invaluable in addressing quality issues and mitigating risks.

A prominent pharmaceutical company faced a recurring issue with contamination in one of its drug products, posing potential health risks and regulatory concerns.

By implementing the 8D corrective action process, a cross-functional team investigated the issue, identifying root causes related to inadequate environmental controls in the manufacturing facility and inconsistencies in the cleaning and sterilization procedures.

Interim containment actions included quarantining and recalling affected product batches, while permanent corrective actions focused on upgrading the facility’s HVAC systems, revising cleaning and sterilization protocols, and implementing enhanced environmental monitoring.

Automotive Industry (origin of 8D Corrective Action)

It is fitting to revisit the automotive industry, where the 8D methodology originated. In a recent case study, a major automaker faced recurring issues with engine failures in one of their popular vehicle models, leading to costly warranty claims and customer dissatisfaction.

By implementing the 8D process, a cross-functional team investigated the issue, identifying root causes related to a design flaw in the engine’s cooling system and inadequate testing procedures during the product development phase.

Interim containment actions included issuing technical service bulletins and providing temporary cooling system modifications for affected vehicles.

Permanent corrective actions focused on redesigning the engine’s cooling system, implementing more rigorous testing protocols, and enhancing communication between the engineering and manufacturing teams.

Through the 8D process and integration with their quality management practices, the automaker successfully resolved the engine failure issue, regained customer trust, and enhanced their overall product quality and reliability.

The 8D corrective action problem-solving method has proven extremely useful for handling thorny quality issues, continuously upgrading workflows, and cultivating an excellence culture in businesses.

By pairing its structured team approach with analytical tools and fact-based choices, the 8D process empowers companies to uncover root causes. It also helps implement lasting fixes and prevent repeating mistakes through establishing protective measures.

As the case studies and examples show, it’s been put to great use across many industries from manufacturing to healthcare where it originated in automotive.

Its flexibility and power have made 8D valued for boosting quality, improving customer satisfaction and staying ahead competitively no matter the market.

The Eight Disciplines methodology remains a strong tool for companies serious about excellence, innovation, and customer focus.

By wholeheartedly embracing this robust framework and blending it with modern quality practices, businesses can expertly handle complex problems. They can also unlock fresh opportunities and build the foundation for sustainable success.

In other words, don’t sleep on 8D corrective action problem-solving. Its fact-based, team-centric transformation approach strengthens any organization now and into the future.

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What is troubleshooting and why is it important?

Troubleshooting is an essential skill used across various fields, including information technology and customer service . Essentially, it involves identifying, diagnosing, and solving problems, as well as understanding why they happened and how to prevent them in the future. This introductory blog post is designed to clarify what troubleshooting really is, presenting it not just as a technical requirement but as a basic method for addressing challenges in everyday life and professional settings. By delving into the nature of troubleshooting, we can recognize its importance and see how it can be applied in different situations.

Understanding Troubleshooting

Troubleshooting is a method of problem-solving widely used to identify, analyze, and resolve issues in various systems, whether they involve technology, business processes, or daily life scenarios. At its core, troubleshooting is about adopting a systematic approach to discover the root cause of a problem and then using knowledge and reasoning to fix it. It involves more than just addressing the immediate issues; it also focuses on understanding why these issues occurred and how similar problems can be avoided or minimized in the future.

The process starts with accurately identifying and defining the problem. This is followed by a detailed analysis of potential causes, where critical thinking and experience play crucial roles. Troubleshooters generate hypotheses about possible faults based on the symptoms they observe. The subsequent steps include testing these hypotheses through experiments or logical deduction and implementing the most effective solution. However, troubleshooting isn’t solely about resolving the issue—it also emphasizes learning from the situation. Effective troubleshooting requires documenting both the problem and its solution, which serves as a valuable reference to prevent future issues. By understanding troubleshooting as a structured yet adaptable approach, individuals and organizations can cultivate a proactive mindset that reduces downtime and enhances efficiency.

What are the symptoms of the troubleshooting?

Symptoms are the clues that you can use to indicate something that is not right with you. Singling out the signs of a problem is just like putting together pieces of a puzzle; every symptom gives a hint as to what’s causing the problem. First of all, the diagnosis of these symptoms correctly is the most important thing that will help to fix the problem.

Here are some common indicators:

• Unusual noises or vibrations
• On the screens or display panels, error messages appear.
• Slow performance or unresponsiveness
• Unexpected shutdowns or restarts
• The structural damages are the most common ones that may include cracks, leaks, or burns.
• Inconsistent results or outputs

How does Troubleshooting Work?

It’s a systematic approach that involves several critical steps to identify and fix issues. Troubleshooting usually follows these steps:

Identify the Problem:

• You start by observing the symptoms: what’s not working, when did it stop working, and how does this differ from normal operation? Gathering information might involve asking users about their recent activities, checking logs, or looking for error messages. It’s all about collecting as much data as possible to understand what the problem is.

Reproduce the Issue:

• Once you have an idea of what might be wrong, try to make it happen again. Reproducing the issue can confirm that you’ve correctly identified the problem and understand the conditions that cause it. This step is crucial because it can help you see if the problem is consistent, random, or triggered by specific actions.

Narrow Down the Causes:

• Now, you begin eliminating potential causes. If you think about a car not starting, you might check if there’s gas in the tank, if the battery is charged, or if the starter is functional. By ruling out what isn’t the cause, you can hone in on what could be. This process may involve checking settings, hardware components, or software configurations.

Find a Solution:

• With a good idea of the root cause, you can now brainstorm potential fixes. Sometimes, the solution is straightforward, like replacing a faulty cable. Other times, it may require more creative problem-solving, like updating software or tweaking settings. If an immediate solution isn’t available, finding a workaround can help you manage the situation temporarily.

Test the Solution:

• After applying a fix, it’s time to see if it works. This means going back to the conditions under which the problem was initially identified and checking to see if the issue has been resolved. Testing is critical to ensure that the solution is effective and doesn’t introduce new problems. It often involves monitoring the fix over time to ensure the problem doesn’t reoccur.

Why is Troubleshooting Important?

Without troubleshooting , we’d just be guessing and trying random things to fix issues, which can be a waste of time and resources. Troubleshooting saves us from that. It’s a systematic approach to solving problems that helps us fix things quickly and efficiently. It’s important because it keeps our daily tools and systems running and prevents small issues from turning into big ones.

• Clear and precise documentation is crucial for boosting productivity. It ensures that, in times of stress when memory might lapse, the troubleshooting personnel can swiftly return to peak performance by consulting the guide.
• The significance of thorough documentation in troubleshooting and beyond underscores that it captures an essential and inventive process centered on solving and preventing problems. It enables teams to tackle customer problems or in-house technical difficulties more rapidly and in a unified manner. It guides both newcomers and clients through the process of fixing issues.
• Well-organized documentation can serve as a treasure trove of knowledge for future reference, eliminate current obstacles, and uphold standards of quality.
• Improved troubleshooting efficiency within a customer service team heightens their confidence and devotion to the company and its brand.

Examples of Troubleshooting

Troubleshooting resources.

Whether you’re dealing with a software glitch, hardware failure, or any other technical hiccup, these tools can help streamline the troubleshooting process and lead you to a solution more efficiently. From detailed guides to interactive forums, the resources available are varied and cater to different levels of expertise and types of problems.

Here are some valuable troubleshooting resources you can turn to:

1. Manufacturer’s Manuals and Support Pages: These documents offer the first line of defense, providing device-specific instructions and solutions.

• Example: https://support.apple.com for Apple products

2. Online Tech Forums: Communities where users and experts discuss problems and solutions can be incredibly helpful.

• Example: https://www.techsupportforum.com

3. FAQs and Knowledge Bases: Many companies offer comprehensive FAQs and knowledge bases with answers to common issues.

• Example: https://support.microsoft.com/en-us

4. Video Tutorials: Platforms like YouTube have countless video tutorials that can provide visual guidance on fixing issues.

• Example: https://www.youtube.com

5. Help Desks and Customer Service: Direct support from trained professionals can provide personalized help.

• Example: Company-specific support lines or help desks

6. Social Media Tech Help Groups: Groups on platforms like Facebook or Reddit can offer advice and support.

• Example: r/techsupport on Reddit

7. DIY Repair Websites: Websites like iFixit provide guides for those looking to fix hardware issues themselves.

• Example: https://www.ifixit.com

Troubleshooting is an essential skill that helps us solve problems in an organized way. Whether it’s a gadget at home, a machine at work, or even a personal project, knowing how to troubleshoot means you can find solutions faster and get things running smoothly again. The next time something goes wrong, remember that troubleshooting is your best friend to get to the bottom of the issue and fix it for good.

What is troubleshooting? – FAQs

What exactly is troubleshooting.

Troubleshooting is a systematic approach to identifying, diagnosing, and solving a problem in a malfunctioning system or piece of equipment. It involves observing symptoms, finding the cause, and implementing a solution.

Why is troubleshooting important?

Troubleshooting is essential because it allows for problems to be fixed efficiently, minimizing downtime and potential damage. It also helps in maintaining the functionality and longevity of equipment or systems.

Can troubleshooting be applied to both hardware and software problems?

Yes, troubleshooting can be applied to any type of problem, whether it’s a physical hardware issue, a software glitch, or even a process-related challenge.

Do I need special skills to troubleshoot effectively?

Effective troubleshooting requires critical thinking, attention to detail, and sometimes specific technical knowledge, but anyone can learn basic troubleshooting techniques for everyday problems.

Is it always necessary to follow the steps of troubleshooting in order?

While it’s generally best to follow the steps in order, experience may allow you to skip steps or tackle them out of sequence if you recognize a familiar problem.

How can I improve my troubleshooting skills?

Practice is key. The more you troubleshoot , the better you’ll become at quickly identifying problems and finding effective solutions. It also helps to learn from others and stay informed about common issues in your area of interest.

What should I do if I can’t solve a problem through troubleshooting?

If troubleshooting doesn’t resolve the issue, it’s wise to consult a professional or seek help from support forums, user manuals, or manufacturer customer service. Sometimes, an outside perspective can provide the solution you need.

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Examples of SMART Goals for Students with Autism

Discover effective examples of SMART goals for students with autism to foster academic and social growth.

Understanding SMART Goals for Students with Autism

In the educational realm, setting clear and attainable objectives is paramount. This is especially crucial for students with autism. One method that has proven effective in this regard is the setting of SMART goals.

Importance of SMART Goals

The acronym SMART stands for Specific, Measurable, Achievable, Relevant, and Time-bound. These goals can be an effective tool for helping students with autism reach their full potential. By setting objectives that are clear, assessable, and within reach, students are more likely to stay motivated and engaged in their learning journey [1].

For example, an academic SMART goal for a student with autism could be to improve communication skills by using complete sentences in classroom discussions. This goal is specific (using complete sentences), measurable (the teacher can track the number of times the student uses complete sentences), achievable (with practice and guidance, the student can learn to use complete sentences), relevant (communication skills are crucial in the classroom and beyond), and time-bound (the student could aim to achieve this goal within a semester).

Benefits of Setting SMART Goals

There are several advantages to setting SMART goals for students with autism. Firstly, these goals provide clear expectations. This can help to reduce anxiety and confusion, common challenges faced by students with autism. Secondly, the measurable nature of SMART goals enables educators and caregivers to objectively assess a student's progress. This can assist in identifying areas where more support or intervention might be needed.

Furthermore, SMART goals can help improve social and communication skills. Goals such as initiating conversations with peers or participating in group activities can help students with autism to better navigate social situations [1]. Similarly, behavioral SMART goals can aid in managing behaviors and enhancing social interactions. This can include using appropriate greetings and practicing turn-taking during conversations.

In conclusion, SMART goals offer a structured and effective approach towards helping students with autism to achieve their potential. By setting and working towards these goals, students can make significant strides in their academic, social, and personal development.

Examples of Academic SMART Goals

When setting academic goals for students with autism, it's crucial to make them SMART - Specific, Measurable, Achievable, Relevant, and Time-bound. This approach can be an effective tool for helping students reach their full potential [1]. Here are some examples of academic SMART goals for students with autism.

Improving Communication Skills

Communication is a key skill in any academic setting. For students with autism, a SMART goal could be improving communication skills by using complete sentences during classroom discussions. This goal is specific, focusing on the use of complete sentences. It's measurable, as educators can track the number of times a student uses complete sentences in discussions. It's achievable and relevant as it directly impacts their ability to communicate effectively in an academic setting. Moreover, the goal can be time-bound, aiming to achieve a certain level of improvement within a given academic term.

Enhancing Reading Comprehension

Reading comprehension is another vital academic skill. A SMART goal in this area could be to understand and summarize the main ideas of a text after reading. This goal is specific and measurable, as it requires the student to demonstrate understanding by summarizing the main ideas. It's achievable with proper support and is indeed relevant to academic success. It can be time-bound by setting a goal for the student to reach a certain level of comprehension within a semester.

Developing Math Problem-Solving Skills

Math problem-solving is a critical skill for students with autism to develop. A SMART goal here could be to solve a certain number of math problems correctly each week. This goal is specific, focusing on solving math problems. It's measurable, as the number of correctly solved problems can be tracked. It's achievable with dedicated practice and is relevant to the student's academic performance in math. Lastly, it's time-bound, aiming to achieve the goal on a weekly basis.

Improving Writing Skills

Writing is another essential academic skill. A SMART goal for improving writing skills could be to write clear and concise sentences in essays or assignments. This goal is specific, focusing on the clarity and conciseness of writing. It's measurable, as educators can assess the clarity and conciseness of the student's writing. It's achievable with practice and is relevant to academic success. This goal can also be time-bound, aiming for improvement within a school year.

These examples of SMART goals for students with autism aim to enhance their academic skills in a structured, measurable way. Remember that the most effective goals are tailored to the individual needs and abilities of each student, and should be adjusted as the student grows and develops.

Examples of Social and Communication SMART Goals

Developing social and communication skills is crucial for students with autism. SMART goals provide a structured framework to help these students enhance their social interactions, build meaningful relationships, and improve their self-confidence. Here are some examples of SMART goals focused on social and communication skills for students with autism.

Initiating Conversations

One of the key areas of social communication that can be challenging for students with autism is initiating conversations. A SMART goal focused on this skill might be: "By the end of the academic year, the student will independently initiate a conversation with peers at least 4 times during a school day, as observed by the classroom teacher."

This goal is specific (initiating conversations), measurable (at least 4 times a day), achievable (with practice and support), relevant (a crucial social skill), and time-bound (by the end of the academic year).

Using Appropriate Nonverbal Cues

Nonverbal communication forms a significant part of our daily interactions. For students with autism, understanding and using appropriate nonverbal cues can be a challenge. A SMART goal for enhancing nonverbal communication skills might be: "Within the academic year, the student will accurately use three appropriate nonverbal communication cues (like eye contact, nodding, and facial expressions) during conversations with peers, as observed by the classroom teacher" [2].

This goal is specific (using nonverbal cues), measurable (three appropriate cues), achievable (with practice and support), relevant (an important aspect of social interaction), and time-bound (within the academic year).

Practicing Turn-Taking in Conversations

Turn-taking in conversations is another important social skill. A SMART goal targeted at this skill might be: "By the end of the semester, the student will wait for his turn to speak during classroom discussions, demonstrating this behavior in 5 out of 5 opportunities, as observed by the classroom teacher."

This goal is specific (waiting for turn to speak), measurable (in 5 out of 5 opportunities), achievable (with practice and reinforcement), relevant (a key communication skill) and time-bound (by the end of the semester).

Through the use of SMART goals, students with autism can make significant strides in their social and communication skills. These examples serve as a starting point for educators and caregivers to tailor goals to the individual needs and abilities of each student. The ultimate goal is to enhance their social interactions and achieve increased self-confidence and independence.

Examples of Behavioral SMART Goals

Setting behavioral SMART goals for students with autism can be a powerful tool for helping them manage their behaviors and improve their social interactions. These goals are specific, measurable, attainable, relevant, and time-bound, and can lead to significant improvements in a student's behavior and social skills [4].

Reducing Disruptive Behaviors

One of the areas where SMART goals can be particularly effective is in reducing disruptive behaviors. For example, a SMART goal could be: "By the end of the academic year, the student will reduce instances of disruptive behaviors, such as interrupting or yelling, by 50% during class time." This goal is specific, measurable (disruptions can be tallied), attainable (with appropriate interventions), relevant (disruptive behavior impacts learning), and time-bound (by the end of the academic year).

Improving Self-Regulation Skills

Another area where SMART goals can be beneficial is in improving self-regulation skills. For instance, a SMART goal could be: "Within six months, the student will demonstrate improved self-regulation skills by utilizing strategies to manage emotions, transitions, and behavioral responses in various settings," as described by ABTaba . This goal is specific (it focuses on self-regulation skills), measurable (the use of strategies can be observed and recorded), attainable (with the right support and practice), relevant (self-regulation is important for success in various settings), and time-bound (to be achieved within six months).

Enhancing Social Interactions

Improving social interactions is another area where SMART goals can be useful. A relevant goal could be: "Within the academic year, the student will enhance nonverbal communication skills by using appropriate gestures, eye contact, facial expressions, and body language in social interactions." This goal, mentioned in ABTaba , is specific (it targets nonverbal communication skills), measurable (use of appropriate nonverbal cues can be observed), attainable (with instruction and practice), relevant (nonverbal communication is crucial for social interactions), and time-bound (to be achieved within the academic year).

These examples of behavioral SMART goals for students with autism demonstrate how this goal-setting approach can be used to target specific areas of need and track progress over time. By setting and working towards these goals, students with autism can make significant strides in managing their behaviors and improving their social interactions.

Implementing SMART Goals Effectively

Implementing SMART goals effectively requires careful thought and detailed planning. By focusing on measurability, tailoring goals to individual needs, and monitoring progress, educators and therapists can help students with autism achieve their goals and improve their skills.

Measurability in Goal Setting

Measurability is a key aspect of SMART goal setting for students with autism. It allows educators to objectively assess a student's progress by tracking specific indicators. For example, educators can track the number of times a student participates in class discussions or their scores on tests. This objective data provides a clear measure of the student's progress and highlights any areas that may need additional focus or intervention [1].

Measurable goals also provide a clear endpoint or target for the student to work towards. This can be motivating for the student and help them see the value of the skills they are learning.

Tailoring Goals to Individual Needs

Each student with autism is unique and will have different needs and abilities. Therefore, it's important that SMART goals are tailored to the individual needs of each student.

For example, SMART goals can help students with autism improve their social and communication skills. Goals could include initiating conversations with peers and participating in group activities. Other students might have goals focused on managing behaviors and improving social interactions, such as using appropriate greetings and taking turns during conversations [1].

By creating personalized goals that are meaningful and relevant to the student, educators can increase the likelihood of the student's engagement and success.

Progress Monitoring and Adjustments

Progress monitoring is an integral part of implementing SMART goals. Regular check-ins can provide valuable information about the student's progress towards their goals, and highlight any areas that may need additional support or intervention.

Additionally, it's important to remember that goals may need to be adjusted over time. If a student is struggling to meet a specific goal, it may be necessary to modify the goal to make it more achievable. On the other hand, if a student is consistently meeting a goal with ease, the goal may need to be made more challenging to continue promoting growth and learning.

Setting and implementing SMART goals is essential for the educational growth and development of students with autism. By focusing on measurability, individual needs, and progress monitoring, educators can create effective learning plans that help students with autism reach their full potential.

[1]: https://www.totalcareaba.com/autism/smart-goals-students-with-autism-examples/

‍ [2]: https://www.abtaba.com/blog/examples-of-smart-goals-for-students-with-autism/

‍ [3]: https://www.adinaaba.com/post/examples-of-smart-goals-for-students-with-autism

‍ [4]: https://www.totalcareaba.com/autism/smart-goals-students-with-autism-examples

Steven Zauderer

CEO of CrossRiverTherapy - a national ABA therapy company based in the USA.

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Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem

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• Published: 07 May 2024
• Volume 57 , article number  135 , ( 2024 )

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• Nezir Aydin 1 , 2 ,
• Alper Murat 3 &
• Boris S. Mordukhovich 4  

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Selecting facility locations requires significant investment to anticipate and prepare for disruptive events like earthquakes, floods, or labor strikes. In practice, location choices account for facility capacities, which often cannot change during disruptions. When a facility fails, demand transfers to others only if spare capacity exists. Thus, capacitated reliable facility location problems (CRFLP) under uncertainty are more complex than uncapacitated versions. To manage uncertainty and decide effectively, stochastic programming (SP) methods are often employed. Two commonly used SP methods are approximation methods, i.e., Sample Average Approximation (SAA), and decomposition methods, i.e., Progressive Hedging Algorithm (PHA). SAA needs large sample sizes for performance guarantee and turn into computationally intractable. On the other hand, PHA, as an exact method for convex problems, suffers from the need to iteratively solve numerous sub-problems which are computationally costly. In this paper, we developed two novel algorithms integrating SAA and PHA for solving the CRFLP under uncertainty. The developed methods are innovative in that they blend the complementary aspects of PHA and SAA in terms of exactness and computational efficiency, respectively. Further, the developed methods are practical in that they allow the specialist to adjust the tradeoff between the exactness and speed of attaining a solution. We present the effectiveness of the developed integrated approaches, Sampling Based Progressive Hedging Algorithm (SBPHA) and Discarding SBPHA (d-SBPHA), over the pure strategies (i.e. SAA). The validation of the methods is demonstrated through two-stage stochastic CRFLP. Promising results are attained for CRFLP, and the method has great potential to be generalized for SP problems.

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1 Introduction

Most real world modeling and optimization problems are subject to uncertainties in problem parameters. Stochastic Programming (SP) methodologies are often resorted for solving such problems either exactly or with a statistical bound on the optimality gap. Beginning with Dantzig ( 1955 ) introduction of a recourse model, the SP models and methodologies has become an important tool for optimization under uncertainty. Operations research community’s contributions to SP has been steadily increasing in terms of both modeling and solution approaches (Lulli and Sen 2006 ; Topaloglou et al. 2008 ; Bomze et al. 2010 ; Peng et al. 2011 ; Toso and Alem 2014 ; Shishebori and Babadi 2015 ; Crainic et al. 2016 ; Celik et al. 2016 ; Aydin 2016 ; Cheung and Simchi-Levi 2019 ; Zhao et al. 2019 ; Juan et al. 2021 ; Jamali et al. 2021 ; Sener and Feyzioglu 2022 ). A common assumption in applying most of SP methodologies is that the probability distributions of the random events are known or can be estimated with an acceptable accuracy. For majority of the SP problems, the objective is to identify a feasible solution that minimizes or maximizes the expected value of a function over all possible realizations of the random events (Solak 2007 ).

In stochastic problems, decomposition methods usually studied into two groups: “stage-based methods and scenario-based methods” (Guo et al. 2015 ). For stage-based decomposition, L-shaped method, or Benders’ decomposition, can be counted. Benders’ decomposition can be used if the first stage decision variables are integer and the second stage variables are real variables (Sen 2005 ; Kim and Zavala 2018 ). Nevertheless, since the Benders’ decomposition depend on strong duality, it cannot be applied in the circumstances of general mixed-integer stochastic programs (Fakhri et al. 2017 ; Li and Grossmann 2018 ). On the other hand, as a scenario-based decomposition method, PHA is motivated by the augmented Lagrangean method, and Gade et al. ( 2016 ) showed that a lower bound can be acquired by PHA, which is like solving the Lagrangean subproblems. Though the PHA is only guaranteed to converge for problems that have continuous and convex (Rockafellar and Wets 1991 ) second stage, Watson and Woodruff ( 2011 ) proposed various heuristics to hasten the convergence of PHA for stochastic mixed-integer programs. So, in PHA none of the subproblems need to be solved to optimality and PHA can discover high-quality solutions within a reasonable amount of time (Guo et al. 2015 ). As stated in Guo et al. ( 2015 ) one of the advantages of scenario-based decomposition methods (as in PHA) over the stage-based ones, i.e., L-shaped or Benders’ decomposition, is their easing of the computational complexity. For the reasons given, we are motivated to, in this study, marry PHA and SAA to get promising results within a reasonable amount of computational time.

In this study, we propose a novel approach, called Sampling Based Progressive Hedging Algorithm (SBPHA) and an improved version of SBPHA (discarding-SBPHA) to solve a class of two-stage SP problems. A standard formulation of the two-stage stochastic program is (Kall and Wallace 1994 ; Birge and Louveaux 1997 ; Ahmed and Shapiro 2002 ):

is the optimal value and \(\xi := \left( f,T,W,h \right)\) denotes the vector of parameters of the second stage problem. It is assumed that some or all of the components of \(\xi\) are random. The expectation term in (1) is then taken with respect to a known probability distribution of \(\xi\) . The problem (1) solves for the first stage variables, \(x \in R^{n_{1}},\) which must be decided prior to any realization of \(\xi ,\) and problem (2) solves for the second stage variables, \(y \in R^{n_{2}}\) , given the first stage decisions as well as the realization of \(\xi\) .

The proposed SBPHA approach hybridizes the Progressive Hedging Algorithm (PHA), and the external sampling based approximation algorithm, Sample Average Approximation (SAA), to efficiently solve a class of two-stage SP problems. While the standard SAA procedure is effective with sufficiently large samples, the required sample size can be quite large for the desired confidence level. Furthermore, for combinatorial problems where the computational complexity increases faster than linearly in the sample size, SAA is often executed with a smaller sample size by generating and solving several SAA problems with i.i.d. samples. For complexity and modeling under uncertainty, readers are encouraged to follow the work conducted by Weber et al. ( 2011 ). These additional SAA replications with the same sample size are likely to provide a better solution than the best solution found so far. However, by selecting the best performing sample solution, the SAA procedure effectively discards the remaining sample solutions which contain valuable information about the problem’s uncertainty. The main idea of the proposed hybrid method SBPHA is to re-use all the information embedded in sample solutions by iteratively solving the samples with adding augmented Lagrangian penalty term (as in PHA) to find a common solution that all samples agree on.

As one of the strategic decision, facility locations require substantial investments to anticipate for uncertain future events (Owen and Daskin 1998 ; Melo et al. 2009 ), i.e., disruption of facilities that are critical to satisfy customer demand (Schütz et al. 2009 ). For a detailed review of uncertainty considerations in facility location problems, readers are referred to Snyder ( 2006 ) and Snyder and Daskin ( 2005 ). Approximation algorithms that are generated for UFLP cannot be practical to the general class of facility location problems such as Capacitated Reliable Facility Location Problem (CRFLP). In practice, capacity decisions are considered jointly with the location decisions. Further, the capacities of facilities often cannot be changed (or at a reasonable cost) in the event of a disruption. Following a facility failure, customers can be assigned to other facilities only if these facilities have sufficient available capacity. Thus, CRFLPs are more complex than their uncapacitated counterparts (Shen et al. 2011 ). Thus, in this study, we introduce promising algorithms for CRFLP, which can be applied to other types of facility location problems and even ex- tended for other two-stage SP problems.

Specifically, the main contributions of the study are as follows: The proposed SBPHA provides a configurable solution procedure, which increases the sampling-based methods’ accuracy and PHA’s efficacy, for a class of two-stage SP problems. Further, the augmented SBPHA which is called Discarding-SBPHA (d-SBPHA), is developed for SPs with binary first stage decision variables. It is analytically proved that SBPHA guarantee optimal solution to the mentioned problems when the number of discarding iterations approaches to infinity.

The rest of the paper is organized as follows. In Sect. 2 , related literature is summarized. In Sect. 3 , we briefly summarize the SAA and PHA methods, and then describe the Sampling Based Progressive Hedging Algorithm (SBPHA) and d-SBPHA in detail. In Sect. 4 , we present scenario based capacitated reliable facility location problem and its mathematical formulation, and then report the computational experiments comparing the solution quality and CPU time efficiency of the SAA algorithm and the proposed hybrid algorithms (SBPHA and d- SBPHA). We conclude with discussions, limitations and future research directions in Sect. 5 .

2 Related literature

SP plays a substantial role in the analysis, design, and operation of systems that includes uncertainties. Further, algorithms developed for SP problems provide a tool for coping with inherent system uncertainty (Aydin 2022 ). The main reason for using SP is that, in general, the SP can successfully accommodate decision making practices under uncertainty. On the other hand, SP with recourse can take curative decisions once the uncertainty is uncovered, and among the SP approaches with recourse, the most commonly used one is the two-stage SP (Ning and You 2019 ).

Accordingly, this study contributes to the solution methodologies for two-stage capacitated reliable facility location problem (CRFLP) in SPs. In the two-stage SP problems, the decision variables are partitioned into two main sets where the first stage variables correspond to decisions prior to the realization of uncertain parameters. Once the random events realized, design or operative strategy improvements (i.e., second stage recourse decisions) can be made at a certain cost. The objective is to optimize the sum of first stage costs and the expected value of the random second stage or recourse costs (Ahmed and Shapiro 2002 ). An extensive number of solution methods have been proposed for solving two-stage SP problems. These solution methods can be classified as either exact or approximation methods (Schneider and Kirkpatrick 2007 ). Exact solution methods include both analytical solution methods as well as computational algorithms. Because of complexity of SP, several approximation algorithms in the form of sampling-based methods (Ahmed and Shapiro 2002 ) or approximation methods (Higle and Sen 1991 ) are proposed.

When the random variable set is finite with a relatively small number of joint realizations (i.e., scenarios), a SP can be formulated as a deterministic equivalent program and solved “exactly" via an optimization algorithm (Rockafellar and Wets 1991 ). The special structure of this deterministic equivalent program also allows for the application of large-scale optimization techniques, e.g., decomposition methods. Such decomposition methods can be categorized into two types. The first type decomposes the problem by stages e.g., L-shaped method (Slyke and Wets 1969 ; Birge and Louveaux 1997 ; Elçi and Hooker 2022 ) while the second type decomposes the problem by scenarios. The latter category’s methods are primarily based on Lagrangian relaxation of the non-anticipativity constraints where each scenario in the scenario tree corresponds to a single deterministic mathematical program e.g., Progressive Hedging Algorithm (Rockafellar and Wets 1991 ; Lokketangen and Woodruff 1996 ; Rockafellar 2019 ). A subproblem obtained by stage or scenario composition may include multiple stages in “by-stage" decomposition method and multiple scenarios in “by-scenario" decomposition methods, respectively (Chiralaksanakul 2003 ). Monte Carlo sampling based algorithms, a popular approximation method, are commonly used in solving large scale SP problems (Morton and Popova 2001 ). Monte Carlo sampling method can be deployed within either ‘interior’ or ‘exterior’ of the optimization algorithm. In the ‘interior’ sampling based methods the computationally expensive or difficult exact computations are replaced with the Monte Carlo estimates during the algorithm execution (Verweij et al. 2003 ). In the ‘exterior’ sampling based methods, the stochastic process is approximated by a finite scenario tree obtained through the Monte Carlo sampling. The solution to the problem with the constructed scenario tree is an approximation of the optimal objective function value. The ‘exterior’ sampling-based method is also referred to as the “sample average approximation" method in the SP literature (Shapiro 2002 ).

The SAA method has become a popular technique in solving large-scale SP problems over the past decade due to its application ease and scope. It has been shown that the solutions obtained by the SAA converge to the optimal solution as the sample size is sufficiently large (Ahmed and Shapiro 2002 ). However, these sample sizes could be quite large and the actual rate of convergence de- pends on the problem conditioning. Several studies reported successful appli- cations of SAA to various stochastic programs (Verweij et al. 2003 ; Kleywegt et al. 2002 ; Shapiro and Homem-de-Mello, 2001 ; Fliege and Xu 2011 ; Wang et al. 2011 ; Long et al. 2012 ; Hu et al. 2012 ; Wang et al. 2012 ; Aydin and Murat 2013 ; Ayvaz et al. 2015 ; Bidhandi and Patrick 2017 ; Lam and Zhou 2017 ; Bertsimas et al. 2018 ; Cheng et al. 2019 ; Cheung and Simchi-Levi 2019 ). Most SP problems have key discrete decision variables in one or more of the 5 stages, e.g., binary decisions to open a plant (Watson and Woodruff 2011 ). Rockafellar and Wets ( 1991 ) proposed a scenario decomposition-based method (PHA) to solve challenging large linear or mixed-integer SP problems. PHA is particularly suitable for SPs where there exists effective techniques for solving individual scenario subproblems. While the PHA possesses theoretical con- vergence properties for SP problems with continuous decisions, it is used as a heuristic method when some or all decision variables are discrete (Lokketangen and Woodruff 1996 ; Fan and Liu 2010 ; Watson and Woodruff 2011 ). There are many application studies of the progressive hedging approach, i.e., lot sizing (Haugen et al. 2001 ), portfolio optimization (Barro and Canestrelli 2005 ), resource allocation in network flow (Watson and Woodruff 2011 ), operation planning (Gonçalves et al. 2012 ), forest planning (Veliz et al. 2015 ), facility location (Gade et al. 2014 ), server location and unit commitment (Guo et al. 2015 ), hydro-thermal scheduling (Helseth 2016 ), and server location (Atakan and Sen 2018 ), and resilience-oriented design (Ma et al. 2018 ).

The main reason and motivation to hybridize SAA and PHA originates from the final step of the SAA, where after selecting the best performing solution, the rest of the sample solutions are abandoned. Discarding of all solutions, except the reported one results in the loss of valuable sample information as well as the effort spent in solving each sample. The proposed SBPHA offers a solution to these losses by considering each sample SAA problem as though it is a scenario subproblem in the PHA. Accordingly, in the proposed SBPHA approach, we modify the SAA method by iteratively re-solving the sample SAA problems while, at the end of each iteration, penalizing deviations from the probability weighted solution of the samples and the best performing solution to the original problem (i.e., as in the PHA). Hence, a single iteration of the SBPHA corresponds to the classical implementation of SAA method. Considering this harmony and fit between SAA and PHA, in this study we hybridized these two methods.

3 Solution methodology

In this section we first summarize PHA and SAA methods. Next, we describe proposed algorithms (SBPHA and d-SBPHA) in detail.

3.1 Progressive hedging algorithm

PHA is proposed by Rockafellar and Wets ( 1991 ) as a scenario decomposition-based method to solve challenging large linear or mixed-integer SP problems. A standard approach for solving two-stage SP (1)-(2) is by constructing scenario tree by generating a finite number of joint realizations \(\xi _{s}\) , \(\forall {s} \in {S},\) called scenarios , and allocating to each \(\xi _{s}\) , a positive weight \(p_{s}\) , such that \(\sum _{{s} \in {S}}^{}p_{s}=1\) (Shapiro 2008 ). The generated set, \(\{ \xi _{1}, \ldots , \xi _{|S|} \}\) , of scenarios, with the corresponding probabilities \(p_{s}, \ldots ,p_{|S|},\) is considered a representation of the underlying joint probability distribution of the random parameters. Using this representation, the expected value of the second stage problem’s objective can be calculated as \(E \left[ \varphi \left( x, \xi \right) \right] = \sum _{{s} \in {S}}^{}p_{s} \varphi \left( x, \xi _{s} \right)\) . By duplicating the second stage decisions, \(y_{s}\) , for every scenario \(\xi _{s}\) , i.e. \(y_{s}=y \left( \xi _{s} \right) , \forall {s} \in {S}\) , the two-stage problem (1)-(2) can be equivalently formulated as follows:

where \(\xi _{s} \left( f_{s},T_{s},W_{s},h_{s} \right) ,\forall {s} \in {S}\) , are the corresponding scenarios and \(\textit{G} \left( x,{{\xi }}_{\text{s}}\right)\) represents the feasible region for scenario s defined by constraints in problem (2) (Shapiro 2008 ). To simplify the notation hereafter, we re-express the above extensive form formulation as follows:

where x , first stage decision vector is a common decision vector independent of scenarios, i.e., \({ x_{s}=x, \forall s \in S}\) . The \({ y_{s}}\) represent second stage decision variables which are determined given a first stage decision and a specific \({ \xi _{s}}\) .

Next, we present the pseudo-code of PHA to show how the PHA converges to a common solution taking all scenarios, which belong to the original problem, into account. Let \(\rho\) be a penalty factor \({( \rho >0)}\) , \({\epsilon }\) be a convergence threshold over the first stage decisions and k be the iteration number. The basic PHA Algorithm is stated as follows as presented in Watson and Woodruff ( 2011 ):

PHA Algorithm:

\({k}{:=}{0}\)

For all \({s}{\in }{S},{\ x}^{k}_{s}{:=} \text { argmin} _{{x},{{y}}_{{s}}}\left( {\text{cx}}{+}{{\text{f}}}_{{\text{s}}}{{y}}_{{s}}\right) {:(}{x},{{y}}_{{s}}{)}{\in }{\varphi }\left( {\text{x}},{{{\xi }}}_{{\text{s}}}\right)\)

\({\overline{ {x}}}^{ {k}} {:=}\sum _{ {s} {\in } {S}}{{ {p}}_{ {s}}{ {x}}^{ {k}}_{ {s}}}\)

For all \({s} {\in } {S} {,\ }{ {w}}^{ {k}}_{ {s}} {:=} {\rho }\left( { {x}}^{ {k}}_{ {s}} {-}{\overline{ {x}}}^{ {k}}\right)\)

\({k} {:=} {k} {+} {1}\)

For all \({s} {\in } {S},\)

\({ {x}}^{ {k}}_{ {s}} {:=} \text { argmin}_{ {x},{ {y}}_{ {s}}}\left( {\text{cx}} {+}{{ {\omega }}^{ {k} {-} {1}}_{ {s}} {x} {+}\frac{ {\rho }}{ {2}}{\left\| {x} {-}{\overline{ {x}}}^{ {k} {-} {1}}\right\| }^{ {2}} {+} {\text{f}}}_{ {\text{s}}}{ {y}}_{ {s}}\right) {:(} {x},{ {y}}_{ {s}} {)} {\in } {\varphi }\left( {\text{x}},{ {{\xi }}}_{ {\text{s}}}\right)\)

For all \({s} {\in } {S} {,\ }{ {\omega }}^{ {k}}_{ {s}} {:=}{ {\omega }}^{ {k} {-} {1}}_{ {s}} {+} {\rho }\left( { {x}}^{ {k}}_{ {s}} {-}{\overline{ {x}}}^{ {k}}\right)\)

\({ {\pi }}^{ {k}} {:=}\sum _{ {s} {\in } {S}}{{ {p}}_{ {s}}\left\| { {x}}^{ {k}}_{ {s}} {-}{\overline{ {x}}}^{ {k}}\right\| }\)

If \({ {\pi }}^{ {k}} {>} {\epsilon } {,\ }\) then go to Step 5, else terminate.

When x is continuous, the PHA’s convergence to a common solution \({\bar{x}}\) on which all scenarios agree is linear. However problem becomes much more difficult to solve when x vector is integer due to the non-convexity (Watson and Woodruff 2011 ). Detailed information on the behavior of the PHA methodology can be found in Wallace and Helgason ( 1991 ), Mulvey and Vladimirou ( 1991 ), Lokketangen and Woodruff ( 1996 ), Crainic et al. ( 2011 ), Watson and Woodruff ( 2011 ), Gade et al. ( 2016 ), Barnett et al. ( 2017 ), Atakan and Sen ( 2018 ), Rockafellar ( 2018 ), Sun et al. ( 2019 ), and Rockafellar and Sun ( 2019 ).

3.2 Sample average approximation

Over the past decade SAA method has become a very popular technique in solving large-scale SP problems. The key idea of SAA approach in solving SP can be described as follows. A sample \(\xi _{1}, \ldots , \xi _{N}\) of N realizations of the random vector \(\xi\) is randomly generated, and subsequently the expected value function \(E \left[ \varphi \left( x, \xi \right) \right]\) is approximated by the sample average function \(N^{-1} \sum _{n=1}^{N} \varphi \left( x, \xi \right)\) . In order to reduce variance within SAA, Latin Hypercube Sampling (LHS) may be used instead of uniform sampling. Performance comparisons of LHS and uniform sampling, within SAA scheme, are analyzed in Ahmed and Shapiro ( 2002 ). The resulting SAA problem \(Min_{x \in X} \{ \hat{g}_{N} \left( x \right) :=cx+ N^{-1} \sum _{n=1}^{N} \varphi \left( x, \xi _{n} \right) \}\) is then solved by deterministic optimization algorithms. As N increases, the SAA algorithm’s converges to the optimal solution of the SP in (1) as shown in Ahmed and Shapiro ( 2002 ) and in Kleywegt et al. ( 2002 ). Since solving SAA becomes a challenge with large N, the practical implementation of SAA often features multiple replications of the sampling, solving each of the sample SAA problems, and selecting the best found solution upon evaluating the solution quality using either the original scenario set or a reference scenario sample set. We now provide a description of the SAA procedure as below.

SAA Procedure:

Initialize : Generate M independent random samples \({m=1,2, \ldots ,{M}}\) with scenario sets \({N_{m}}\) where \(\vert {N_{m}} \vert ={N}\) . Each sample m consists of N realizations of independently and identically distributed (i.i.d.) random scenarios. We also select a reference sample \({N^{'}}\) which is sufficiently large, e.g., \(\vert {N^{'} \vert \gg N}\) .

Step 1 : For each sample m , solve the following two-stage SP problem and record the sample optimal objective function value \({v^{m}}\) and the sample optimal solution \({x^{m}}\) .

Step 2: Calculate the average \({\bar{v}^{M}}\) of the sample optimal objective function values obtained in Step 1 as follows:

Step 3: Estimate the true objective function value \({\hat{v}^{m}}\) of the original problem for each sample’s optimal solution. Solve the following problem for each sample using the optimal first stage decisions \({x^{m}}\) from step 1.

Step 4: Select the solution \({x^{m}}\) with the best \({\hat{v}^{m}}\) as \({x^{SAA}}\) as the solution and \({v^{SAA}= {\hat{v}^{m}}}\) as the objective function value of SAA.

Let \({v^{*}}\) denote the optimal objective function value of the original problem (1-2). The \({\bar{v}^{M}}\) is an unbiased estimator of the expected optimal objective function value of sample problems. Hence \({\bar{v}^{M}}\) provides a statistical lower bound on the \({v^{*}}\) (Ahmed and Shapiro 2002 ).When the first and the second stage decision variables in (1) and (2) are continuous, it has been proved that an optimal solution of the SAA problem provides optimal solution of the true problem with probability approaching to one at an exponential rate as N increases (Shapiro and Homem-de-Mello 2001 ; Ahmed and Shapiro 2002 ; Meng and Xu 2006 ; Liu and Zhang 2012 ; Xu and Zhang 2013 ; Shapiro and Dentcheva 2014 ). Determining the optimal or the minimum required sample size, N , is an important task for SAA applicants, therefore a numerous reserachers contributed on determining the sample size (Kleywegt et al. 2002 ; Ahmed and Shapiro 2002 ; Shapiro 2002 ; Ruszczynski and Shapiro 2003a ).

3.3 Sampling based progressive hedging algorithm (SBPHA)

We now describe the proposed SBPHA algorithm which is a hybridization of the SAA and PHA. The motivation for this hybridization originates from the final stage of the SAA method (Step 4, in SAA) where, after selecting the best performing solution, the rest of the sample solutions are discarded. This discarding of the \(\left( M-1 \right)\) sample solutions results in losses in terms of both valuable sample information (increasing with M ) as well as the effort spent in solving for each sample’s solution (increasing with N) . The proposed SBPHA offers a solution to these losses by considering each sample SAA problem as though it is a scenario subproblem in the PHA. Accordingly, in the proposed SBPHA approach, we modify the SAA method by iteratively re-solving the sample SAA problems while, at the end of each iteration, penalizing deviations from the probability weighted solution of the samples and the best performing solution to the original problem (i.e., as in the PHA). Hence, a single iteration of the SBPHA corresponds to the classical implementation of SAA method.

By integrating elements of both SAA and PHA, SBPHA inherits the PHA’s ability to handle problems with Mixed-Integer Programming (MIP) second-stage subproblems. This hybridization allows us to leverage the computational advantages of SAA while addressing the challenges posed by MIP subproblems, resulting in a versatile and efficient algorithm for solving complex stochastic programming problems. An important distinction of the SBPHA from classical PHA is the sampling concept and the size of the subproblems solved. The classical PHA solves many subproblems each corresponding to a single scenario in the entire scenario set one by one at every iteration, and evaluates the probability weighted solution using the individual scenario probabilities. In comparison, the SBPHA solves only a few subproblems each corresponding to the samples (with multiple scenarios) and determines the probability weighted solution in a different way than PHA (explained in detail below). Note that while solving individual sample problems in SBPHA is more difficult than solving a single scenario subproblem in PHA, SBPHA solves much fewer subproblems. Clearly, SBPHA makes a trade-off between the number of sample subproblems to solve and the size of each sample subproblem.

We first present the proposed SBPHA algorithm and then describe its steps in detail. For clarity, we give the notation used precedes the algorithmic steps. Note that for brevity, we only define those notation that are new or different than those used in the preceding sections:

\({k},{ {k}}_{ {max}}\) : iteration index and maximum number of iterations

\({ {P}}_{ {m}},{\widehat{ {P}}}_{ {m}}\) : probability and normalized probability of sample m realization, i.e., \(\sum _{{m}}{{\widehat{ {P}}}_{ {m}} =1}\)

\({ {x}}^{ {m}, {k}} {\ }\) : solution vector for sample m at iteration k

\({\overline{ {x}}}^{ {k}}\) : probability weighted solution vector at iteration k

\({\overline{\overline{ {x}}}}^{ {k}}\) : balanced solution vector at iteration k

\({ {x}}_{ {best}}\) : best incumbent solution

\({\widehat{ {v}}}_{ {best}}\) : objective function value of the best incumbent solution with respect to \({N}'\)

\({\widehat{ {v}}}^{ {k}}_{ {best}}\) : objective function value of the best solution at iteration k with respect to \({N}'\)

\({ {\omega }}^{ {k}}_{ {m}}\) : dual variable vector for sample m at iteration k

\({ {\rho }}^{ {k}} {\ }\) : penalty factor at iteration k

\({\beta }\) : update parameter for the penalty factor, \({1} {<\ } {\beta } {<} {2}\)

\({ {\alpha }}^{ {k}}\) : weight for the best incumbent solution at iteration \({k}, {0} {\le } {\ } {\alpha } {\le } {1}\)

\({ {\Delta }}_{ {\alpha }}\) : update factor for the weight of the best incumbent solution, \({0} {\le } {\ }{ {\Delta }}_{ {\alpha }} {\le } {0}. {05}\)

\({ {\epsilon }}_{ {\alpha }} {\ }\) : Euclidean norm distance between sample solutions \({ {\text{x}}}^{ {m}, {k}}\) and \({\overline{\overline{ {x}}}}^{ {k}}\) at iteration k

\({\epsilon }\) : convergence threshold for solution spread

\({ {x}}^{ {SBPHA}}\) : best solution found by SBPHA

\({ {v}}^{ {SBPHA}}\) : objective function value of the best solution found by SBPHA

The pseudo-code for the Sampling Based Progressive Hedging Algorithm is as follows:

The first step in SBPHA’s initialization is to execute the standard SAA procedure (Step 5-6). In the initialization step of the SBPHA, unlike SAA, we also calculate sample m ’s probability and normalized probabilities, e.g., \({\text{P}}_{\text{m}}\) and \({\widehat{\text{P}}}_{\text{m}}\) , which are used to calculate sample m’s probability weighted average solution \({\overline{\text{x}}}^{\text{k}}\) at iteration \(\text{k}\) (Step 12). Next, in Step 13, we calculate the samples’ balanced solution ( \({\overline{\overline{\text{x}}}}^{\text{k}}\) ) as a weighted average of the average solution ( \({\overline{\text{x}}}^{\text{k}}\) ) and the incumbent best solution ( \({\text{x}}_{\text{best}}\) ). The \({\text{x}}_{\text{best}}\) is initially obtained as the solution to the SAA problem (Step 6) and later on, updated based on the evaluation of the improved sample solutions and the most recent incumbent best (Step 26). In calculating the balanced solution ( \({\overline{\overline{\text{x}}}}^{\text{k}}\) ), the SBPHA uses a weight factor \({{\alpha }}^{\text{k}}{\in }\rm {(0,1)}\) to tune the bias between the sample’s current iteration average solution and the best incumbent solution. High values of \({{\alpha }}^{\text{k}}\) tend the balanced solution (hence the sample solutions in the next iteration) to the samples’ average solution, whereas low values tend \({\overline{\overline{\text{x}}}}^{\text{k}}\) to the incumbent best solution. There are two alternative implementations of SBPHA concerning this bias tuning, where the \({{\alpha }}^{\text{k}}\) can be static by setting \({{\Delta }}_{{\alpha }}\rm {=0}\) or dynamically changing over the iterations by setting \({{\Delta }}_{{\alpha }}\rm {>0}\) (see Step 14). The advantage of dynamic \({{\alpha }}^{\text{k}}\) is that, beginning with a large \({{\alpha }}^{\text{k}}\rm {,\ }\) we first prioritize the sample average solution until the incumbent best solution quality improves. This approach allows guiding the sample solutions to a consensus sample average initially and then directing the consensus sample average in the direction of evolving best solution.

In Step 15, we update the penalty factor \({{\rho }}^{\text{k}}\) depending whether the distance ( \({{\epsilon }}_{\text{k}}\) ) of the sample solutions from the most recent balanced solution has sufficiently improved. We choose the improvement threshold as half of the distance in the previous iteration (e.g., \({{\epsilon }}_{\rm {k-1}}\) ) . Similarly, in Step 16, we update the dual variable ( \({{\omega }}^{\text{k}}_{\text{m}}\) ) using the standard subgradient method. Note that the \(\ {{\omega }}^{\text{k}}_{\text{m}}\) are the lagrange multipliers associated with the equivalence of each sample’s solution to the balanced solution.

In Step 18, we solve each sample problem with additional objective function terms representing the dual variables and calculate the deviation of the sample solutions from the balanced solution (i.e., \({\ }{ {{\epsilon }}}_{ {\text{k}}}\) ). Step 22 estimates the objective function value of each sample solution in the original problem using the reference set \({\ } {\text{N}} {\rm {'}}\) . Steps 25 and 26 identify the sample solution \({ {\text{x}}}^{ {\text{m}}, {\text{k}}}\) with the best \({\widehat{ {\text{v}}}}^{ {\text{m}}, {\text{k}}}\) in iteration \({\text{k}}\) and updates the incumbent best \({\widehat{ {\text{v}}}}_{ {\text{best}}}\) if there is any improvement. Note that \({\widehat{ {\text{v}}}}_{ {\text{best}}}\) is monotonicaly nonincreasing with the SBPHA iterations. The Steps 22, and 25-26 correspond to the integration of SAA method’s selection of the best performing sample solution. Rather than terminating with the best sample solution, the proposed SBPHA conveys this information in the next iteration through the balanced solution. The SBPHA algorithm terminates when either of the two stopping conditions are met. If the iteration limit is reached \({\text{k}} {{\ge }}{ {\text{k}}}_{ {\text{max}}}\) or when the all sample solutions converged to the balanced solution within a tolerance then the SBPHA terminates with the best found solution. The worst-case solution of the SBPHA is equivalent to the SAA solution. This can be observed by noting that the best incumbent solution is initialized with the SAA’s solution and \({\widehat{ {\text{v}}}}_{ {\text{best}}}\) is monotonicaly nonincreasing with the SBPHA iterations. Hence, the SBPHA ensures that there is always a feasible solution which has same performance or better than that of SAA’s. Flow chart of SBPHA algorithm is provided in Fig.  4 .

3.4 Discarding-SBPHA (d-SBPHA) for binary-first stage SP problems

The Discarding-SBPHA extends the SBPHA by finding an improved, and ideally the optimal solution, to the original problem. The main idea of the d-SBPHA is to re-solve SBPHA, by adding constraint(s) to the sample subproblems in (8) that prohibits finding the same best incumbent solution(s) found in earlier d-SBPHA iterations. This prohibition is achieved through constraints that are violated if \({ {\text{x}}}^{ {\text{m}}, {\text{k}}}\) overlaps with any of the best incumbent solutions ( \({ {x}}_{ {best}} {)}\) found thus far in the d-SBPHA iterations. This modification of SBPHA can be considered as globalization of the SBPHA in that by repeating the discarding steps, the d-SBPHA is guaranteed to find the optimum solution, albeit, with infinite number of discarding steps. The d-SBPHA initializes with the \({ {x}}_{ {best}}\) of the SBPHA solution and the SBPHA iteration’s parameter values ( \(\omega\) , \(\alpha\) , \(\rho\) \({\dots }\) ) where this solution is first encountered in the SBPHA iteration history. We now provide the additional notation and algorithmic steps of the d-SBPHA and then describe in detail.

Additional notations for d-SBPHA

\({\text{o}}\) : iteration number where the SBPHA or d- SBPHA finds the solution, where it is found the very first, which is converged at the end,

\({ {\text{d}} {\rm {,\ }} {\text{d}}}_{ {\text{max}}}\) : number discarding iterations and maximum number of discarding iteration,

\({\text{D}}\) : set of discarded solutions,

\({ {\text{n}}}_{{ {\text{D}}}_{ {\text{t}}}}\) : number of binary decision variables that are equal to 1 in discarded solution \({ {\text{D}}}_{ {\text{t}}} {{\in }} {D},\)

\({ {\text{D}}}^{ {\text{1}}}_{ {\text{t}}}\) : set of decision variables, that are equal to 1 in discarded solution t,

\({ {\text{D}}}^{ {\text{0}}}_{ {\text{t}}}\) : set of decision variables, that are equal to 0 in discarded solution t.

Initialization step of the d-SBPHA is the implementation of the original SBPHA with the only difference is to set up the starting values of parameters as the values of the SBPHA algorithm where the current best solution is found. Also in Step 1, the set of the solutions that will be discarded is updated to prevent the algorithm to re-converge to the same solution. In step 2-7, the parameters are updated. Step 8 updates the set of discarded solution by including the most recent \({{\text{x}}}_{{\text{best}}}\) . Step 9 executes SBPHA steps to update the parameters of sample problems. Note that Step 11 has the same objective function as in step 13 of SBPHA with the additional discarding constraint(s) that prevent finding the first-stage solutions that are already found in preceding d-SBPHA iterations. Step 18 executes steps of the SBPHA, which test solutions’ quality and performs the updating of the best solution according to the SAA approach. The only difference, in Step 15, between d-SBPHA and SBPHA is that d-SBPHA checks whether the maximum number of discards is reached. If the discarding iteration limit is reached, then the algorithm reports the solution with the best performance (in Steps 16 and 17) else continues discarding. Note that with the discarding strategy, d-SBPHA terminates with a better or the same solution as SBPHA and is guaranteed to terminate with optimal solution with infinite number of discarding iterations. Lastly, we note that, with increasing discarding iterations, the discarding constraints. Flow chart of d-SBPHA algorithm is provided in Fig.  5 .

3.4.1 Lower bounds on SBPHA and d-SBPHA

The majority of the computational effort of SBPHA (and d-SBPHA) is spent in solving the sample subproblems as well as evaluating the first stage decisions in the larger reference set. Especially, for combinatorial problems with discrete first-stage decisions, the former effort is much significant than the latter. To improve the computational performance, we propose using a sample specific lower bound used while solving the sample subproblems. The theoretical justification for the proposed lower bound for sample problems is that if the balanced solution does not change, then the solution of the sample problems is non-decreasing. Hence, one can use the previously found solution as the lower bound (due to the Lagrangean duality property). However, if the balanced solution changes, then the lower bound property of the previous solution are not guaranteed and hence the lower bound is removed or a conservative estimate of the lower bound is utilized.

Let \({\text{l}}{ {\text{b}}}^{ {\text{m}} {\rm {,\ }} {\text{k}}}\) be the lower bound for sample m at iteration k in SBPHA (or d-SBPHA). In Step 18 in SBPHA (Step 16 in d-SBPHA) a valid cut should be added: \({ {\text{v}}}^{ {\text{m}}, {\text{k}}} {{\ge }}\) \({\text{l}}{ {\text{b}}}^{ {\text{m}} {\rm {,\ }} {\text{k}}}\) , \({{\forall }} {\text{m}}, {\text{m}} {\rm {=}} {\text{1}} {\rm {,\dots }} {\text{M}}\) , where \({\text{l}}{ {\text{b}}}^{ {\text{m}} {\rm {,\ }} {\text{k}}} {\rm {=}}{ {\text{c}}}_{ {\text{lb}}} {\text{l}}{ {\text{b}}}^{ {\text{m}} {\rm {,\ }} {\text{k}} {\rm {-}} {\text{1}}}\) , and \({\text{0}} {{\le }}{ {\text{c}}}_{ {\text{lb}}} {{\le }} {\text{1}}\) , where \({ {\text{c}}}_{ {\text{lb}}}\) is a tuning parameter to adjust the tightness of the lower bound. However \({{\ }}{ {\text{c}}}_{ {\text{lb}}}\) , should not be close to 1 because it might cause infeasible solutions. There is a trade of on value of \({{\ }}{ {\text{c}}}_{ {\text{lb}}}\) . Higher values might cause either infeasible or sub- optimal solutions, lower values does not provide consistent/tight constraints that should help improving solution time. In this study, we tested multiple values for \({{\ }}{ {\text{c}}}_{ {\text{lb}}}\) , we suggest applicants to choose \({{\ }} {\text{0}}. {\text{4}} {{\le }}{ {\text{c}}}_{ {\text{lb}}} {{\le }} {\text{0}}. {\text{6}}\) range. Providing lower bound to optimization problem saved approximately 10%-15% of the solution time.

3.4.2 Theoretical implications of SBPHA and d-SBPHA

In the view of theoretical aspect, the most important merit of the SBPHA from classical PHA is the sampling concept and the size of the subproblems optimized. The classical PHA optimizes numerous subproblems each relating to an individual scenario in the entire scenario set one by one at each iteration. Differently, the SBPHA optimizes only a small number of subproblems each corresponding to the samples (including multiple scenarios) and determines the probability weighted solution in a different way than PHA. Note that optimizing a particular sample problem is more challenging than optimizing a single scenario. On the other hand, the Discarding-SBPHA extends the SBPHA by finding an improved, and ideal optimal solution, to the original problem. The main theoretical contribution of d-SBPHA is to re-solve SBPHA, by adding constraint(s) to the sample subproblems), which prohibits finding the same best incumbent solution(s) found in earlier d-SBPHA iterations. The theoretical characteristics of the SBHA and d-SBHA can be highlighted as follows:

Proposition 1

(Equivalence): SBPHA is equivalent to SAA if algorithm is executed only one time. Further, SBPHA (with parameter settings \({{{\alpha }}}^{{\text{k}}}=1\) , \({{\Delta }}_{{\alpha }}\rm {=0}\) , and \({{\rho }}^{\text{k}}\) is constant for all k), is equivalent to PHA if the samples are mutually exclusive and their union is the entire scenario set.

We show this in two parts.

For SAA: If SBPHA terminated at Step 1, then \({ {\text{x}}}^{ {\text{SBPHA}}} {\rm {=}}{ {\text{x}}}^{ {\text{SAA}}}\) , and \({ {\text{v}}}^{ {\text{SBPHA}}} {\rm {=}} { {\text{v}}}^{ {\text{SAA}}}\) . It can be concluded that SBPHA is equivalent to SAA.

For PHA: Under specified assumptions and for \({\text{M}} {\rm {=}} {\text{S}}\) and \({{\ }}{ {\text{N}}}_{ {\text{m}} {\rm {=}} {\text{1}} {\rm {,\dots ,}} {\text{M}}} {\rm {=}} {\text{1}}\) , SBPHA=PHA.

Let’s consider a two-stage SP problem with finite number of scenarios \({ {{\ }} {{\xi }}}_{ {\text{s}}}\) , \({\text{s}} {\rm {=}} {\text{1}} {\rm {,\dots ,}} {\text{S}}\) , and each scenario occurs with a probability \({ {\text{p}}}_{ {\text{s}}}\) , where \(\sum ^{ {\text{S}}}_{ {\text{s}} {\rm {=}} {\text{1}}}{{ {\text{p}}}_{ {\text{s}}} {\rm {=}} {\text{1}}}\) . Consider SBPHA with samples as the individual scenarios, e.g., \({\text{M}} {\rm {=}} {\text{S}}\) and \({{\ }}{ {\text{N}}}_{ {\text{m}} {\rm {=}} {\text{1}} {\rm {,\dots ,}} {\text{M}} {\rm {,\ }}} {\rm {=}} {\text{1}} {\rm {,\ }}\) where \({{\ \ \ }} {\text{m}} {{\ne }} {\text{m}} {\rm {'}}\) . It can be concluded that \({ {{\ \ }} {\text{P}}}_{ {\text{m}}} {\rm {=}}{ {\text{p}}}_{ {\text{s}}}\) . If weight for the best incumbent solution and update factor for the weight of the best incumbent solution are equal to 1 and 0, consecutively, at every iteration \({\rm {(}}{ {{\alpha }}}^{ {\text{k}}} {{:=}} {\text{1}},{ {{\Delta }}}_{ {{\alpha }}} {\rm {=}} {\text{0}} {\rm {)}}\) , then \({\overline{\overline{ {\text{x}}}}}^{ {\text{k}}} {\rm {:=}}{\overline{ {\text{x}}}}^{ {\text{k}}} {{:=}}\sum _{ {\text{s}} {{\in }} {\text{S}}}{{ {\text{p}}}_{ {\text{s}}}{ {\text{x}}}^{ {\text{k}}}_{ {\text{s}}}}\) , and \({ {\text{x}}}^{ {\text{SBPHA}}} {\rm {=}}{ {\text{x}}}^{ {\text{PHA}}} {{\ }}\) and \({{\ }}{ {\text{v}}}^{ {\text{SBPH}} {\text{A}}} {\rm {=}}{ {\text{v}}}^{ {\text{PHA}}}\) . \(\square\)

Proposition 2

(Convergence): SBPHA algorithm converges and terminates with the best solution encountered.

We prove this by contradiction. Assume that SBPHA finds, at iteration k , the best solution as \(x_{best}=x^{*}.\) Let’s assume that SBPHA algorithm converges to a solution \(x^{'} \ne x_{best}\) which has a worse objective value than \(x^{*}\) (with respect to the reference scenario set). Note that convergence implies \(x^{k}=x^{k-1}=x^{'}\) (assuming \(k_{max}=\infty\) and \(\varepsilon =0\) ). Further in the last update, we must have \(x^{k}=x^{'}= \alpha ^{k}x^{'} + \left( 1- \alpha ^{k} \right) x_{best}\) . Since \(\alpha ^{k}<1\) , this equality is satisfied if and only if \(x^{'}=x_{best}\) which is a contradiction. \(\square\)

Proposition 3

SBPHA and d -SBPHA algorithms have the same convergence properties as SAA with respect to the sample size.

It is showed in Ahmed and Shapiro ( 2002 ) and Ruszczynski and Shapiro ( 2003b ) that SAA converges with probability one (w.p.1) to optimal solution of the original problem as sample size, \(\left( N \rightarrow \infty \right)\) , increases to infinity. Since Step 1 in SBPHA is the implementation of SAA and that SBPHA does converge to the best solution found (Proposition 2), we can simply argue that SBPHA and d-SBPHA converges to optimal solution of the original problem as SAA does with increasing sample size. Furthermore, since SBPHA and d-SBPHA guarantee a better or same solution quality as SAA provides, we can conjecture that SBPHA and d-SBPHA have more chance to reach the optimality than SAA with a given number of samples and sample size. \(\square\)

Proposition 4

The d-SBPHA algorithm converges to the optimum solution as \(d\rightarrow \infty\) .

Given that d-SBPHA is not allowed finding the same solution, in the worst case, the d-SBPHA iterates as many times as the number of feasible solutions (infinite in the continuous and finite in the discrete case) for the first stage decisions before it finds the optimum solution. Note that the proposed algorithm is effective for the problems where the first stage decision variables are binary.

Clearly, as the number of discarding constraints added increases linearly with the number discarding iterations, the resulting problems become more difficult to solve. However, in our experiments for a particular problem type, we observed that, in majority of the experiments, d-SBPHA finds the optimal solution in less than 10 discarding iterations. \(\square\)

3.4.3 Practical aspects of the study

The application areas of SP are vast because of its characteristic in terms of representing real world problems better by considering the uncertainty nature of the environment. Among the SP approaches the most commonly used one is the two-stage SP. The proposed algorithms provide effectiveness in terms of solution quality and efficiency in terms of solution time to two-stage SP problems especially for those that have binary first stage decision variables. Disaster management is one of the fields that includes high level of uncertainty (Rath et al. 2016 ; Caunhye et al. 2016 ). For such problems the proposed SBPHA and d-SBPHA algorithms are very effective tools to apply. Besides, several more applications areas can be counted to apply the proposed algorithms, such as supply chain (Tolooie et al. 2020 ), end-of-life vehicles (Karagoz et al. 2022 ), sharing economy (Aydin et al. 2022 ), workforce capacity planning (Kafali et al. 2022 ), unit commitment (Håberg 2019 ), production planning (Islam et al. 2021 ), power dispatching (Mohseni-Bonab and Rabiee 2017 ), energy (Talari et al. 2018 ), debris management (Deliktaş et al. 2023 ), etc.. Since d-SBPHA provides optimal or near optimal solutions, thanks to its powerful convergence property. Accordingly, d-SBPHA can be applied when decision makers need accurate solutions, and choose to use SBPHA when solution time is more valued than the accuracy.

4 Experimental study

We now describe the experimental study performed to investigate the computational and solution quality performance of the proposed SBPHA and d-SBPHA for solving two-stage SP problems. We benchmark the results of SBPHA and d- SBPHA with those of SAA. All algorithms are implemented in Matlab R2010b and integer programs are solved with CPLEX 12.1. The experiments are conducted on a PC with Intel(R) Core 2 CPU, 2.13 GHz processor and 2.0 GB RAM running on Windows 7 OS. Next, we describe the test problem, CRFLP, in Section 3.1 and experimental design in Section 3.2. In Section 3.3, we report on the sensitivity analysis results of SBPHA and d-SBPHA’s performance with respect to algorithm’s parameters. In Section 3.4, we present and discuss the benchmarking results.

4.1 Capacitated reliable facility location problem (CRFLP)

Facility location is a strategic supply chain decision and require significant investments to anticipate and plan for uncertain future events (Owen and Daskin 1998 ; Melo et al. 2009 ). An example of such uncertain supply chain events is the disruption of facilities that are critical for the ability to efficiently satisfy the customer demand (Schütz et al. 2009 ). These disruptions can be either natural disasters, i.e., earthquake, floods, or man-made, i.e., terrorist attacks (Murali et al. 2012 ), labor strikes. For a detailed review of uncertainty considerations in facility location problems, readers are referred to Arya et al. ( 2004 ), Pandit ( 2004 ), Snyder and Daskin ( 2005 ), Snyder ( 2006 ), Resende and Werneck ( 2006 ), Shen et al. ( 2011 ), Li et al. ( 2013 ), An et al. ( 2014 ), Yun et al. ( 2014 ), Albareda-Sambola et al. ( 2015 ). In practice, capacity decisions are considered jointly with the location decisions. Further, the capacities of facilities often cannot be changed (or at a reasonable cost) in the event of a disruption. Following a facility failure, customers can be assigned to other facilities only if these facilities have sufficient available capacity. Thus capacitated reliable facility location problems are more complex than their uncapacitated counterparts (Shen et al. 2011 ). Gade ( 2007 ) applied the SAA in combination with a dual decomposition method to solve CRFLP. Later, Aydin and Murat ( 2013 ) proposed a swarm intelligence based SAA algorithm to solve CRFLP

We now introduce the notation used for the formulation of CRFLP. Let \(F_R\) , and \(F_U\) denote the set of possible reliable and unreliable facility sites, accordingly, \(F=F_R\bigcup {F_U}\) , denote the set of all possible facility sites, including the emergency facility, and \(\mathcal {D}\) denote the set of customers. Let \(f_i\) be the fixed cost for locating facility \(\ i\in F\) , which is incurred if the facility is opened, and \(d_j\) be the demand for customer \(\ j\in \mathcal {D}\) . The \(\ c_{ij}\) denotes the cost of satisfying each unit demand of customer j from facility i and includes such variable cost drivers as transportation, production, and so forth. There are failure scenarios where the unreliable facilities can fail and become incapacitated to serve any customer demand. In such cases, demand from customers need to be allocated between the surviving facilities and emergency facility ( \({\text{f}}_{\text{e}}\) ) subject to capacity availability. Each unit of demand that is satisfied by the emergency facility cause a large penalty \(\ (h_j)\) cost. This penalty can be incurred due to finding an alternative source or due to the lost sale. Lastly, the facility i has limited capacity and can serve at most \(b_i\) units of demand.

We formulate the CRFLP as a two-stage SP problem. In the first stage, the location decisions are made before the random failures of the located facilities. In the second stage, following the facility failures, the customer-facility assignment decisions are made for every customer given the facilities that have not failed. The goal is to identify the set of facilities to be opened while minimizing the total cost of open facilities and the expected cost of meeting demand of customers from the surviving facilities and the emergency facility. In the scenario based formulation of CRFLP, let s denote a failure scenario and the set of all failure scenarios is \(\ S\) , where \(\ s\in S\) . Let \(p_s\) be the occurrence probability of scenario s and \(\sum _{s\in S}{p_s=1}\) . Further let \(\ k^s_i\) denote whether the facility i survives (i.e., \(k^s_i=1\) ) and \(k^s_i=0\) otherwise. For instance, in the case of independent facility failures, we have \({\left| S\right| =2}^{\left| F_U\right| }\) possible failure scenarios.

The binary decision variable \(x_{i}\) specifies whether facility i is opened, and the variable \(y_{ij}^{s}\) specifies the rate of demand of customer j is satisfied by facility i in scenario s . The scenario based formulation of the CRFLP as a two-stage SP is as follows:

4.2 Experimental setting

We used the test data sets provided in Zhan ( 2007 ) which are also used in Shen et al. ( 2011 ) for URFLP. In these data sets, the coordinates of site locations (facilities, customers) are i.i.d and sampled from \(\text{U}\left[ \text{0,1}\right] {\times U}\left[ \text{0,1}\right]\) . The sets of customer and facility sites are identical. The customer demand is also i.i.d., sampled from \({\ U}\left[ \text{0,1000}\right]\) , and rounded to the nearest integer. The fixed cost of opening an unreliable facility is i.i.d. and sampled from \({\ U}\left[ \text{500,1500}\right]\) , and rounded to the nearest integer. For the reliable facilities, we set the fixed cost to \(\text{2,000}\) for all facilities. The variable costs \({\ }{\text{c}}_{\text{ij}}\) for customer \(\text{j}\) and facility i (excluding the emergency facility) are chosen as the Euclidean distance between sites. We assign a large penalty cost, ( \(\text{20}\) ), \(h_j\) for serving customer \(\text{j}\) from emergency facility. Zhan ( 2007 ) and Shen et al. ( 2011 ) consider URFLP and thus their data sets do not have facility capacities. In all our experiments, we selected identical capacity levels for all facilities, i.e., \({\text{b}}_{\rm {i=1,..,|F|}}\rm {=2,000}\) . Datasets used in our study is given in Appendix B Table  10 .

In generating the failure scenarios, we assume that the facility failures are independently and identically distributed according to the Bernoulli distribution with probability \({{\ q}}_{\text{i}}\) , i.e., the failure probability of facility \(\ \text{i}\) . We experimented with two sets of failure probabilities; first set of experiments consider uniform failure rates, i.e., \({\text{q}}_{\text{i}{\in }{\text{F}}_{\text{U}}}\rm {=q}\) where \(\ \rm {q=}{\{}\rm {0.1,\ 0.2,\ 0.3}{\}}\) , and the second set of experiments consider bounded non-uniform failure rates i.e. \({{\ q}}_{\text{i}}\) , where \({{\ q}}_{\text{i}}{\le }\rm {0.3}\) . We restricted the failure probabilities with \(\rm {0.3}\) since larger failure rates are not realistic. The reliable facilities and emergency facility are perfectly reliable, i.e., \({\ }{\text{q}}_{\text{i}{\in }\rm {(}{\text{F}}_{\text{R}}{\cup }{\text{f}}_{\text{e}}\rm {)}}\rm {=1}\) . Note that in the case, where \(\ {\text{q}}_{\text{i}{\in }{\text{F}}_{\text{U}}}\rm {=0}\) , corresponds to the deterministic fixed-charge facility location problem. The failure scenarios \(\text{s}{\in }\text{S}\) are generated as follows. Let \({\text{F}}^{\text{s}}_{\text{f}}{\subset }{\text{F}}_{\text{U}}\) denote the facilities that are failed, and \({\ }{\text{F}}^{\text{s}}_{\text{r}{{\in }\text{F}}_{\text{U}}}{\equiv }{\text{F}}_{\text{U}}\backslash {\text{F}}^{\text{s}}_{\text{f}}\) be the set of surviving facilities in scenario \({\ s}\) . The facility indicator parameter in scenario \(\text{s}\) become \({\text{k}}^{\text{s}}_{\text{i}}\) =0 if \(\text{i}{\in }{\text{F}}^{\text{s}}_{\text{f}}\) , and \({\text{k}}^{\text{s}}_{\text{i}}\) =1 otherwise, e.g., if \(\text{i}{\in }{\text{F}}^{\text{s}}_{\text{r}}{\cup }{\text{F}}_{\text{R}}{\cup }{\{}{\text{f}}_{\text{e}}\}\) . The probability of scenario \(\text{s}\) is then calculated as \({\ }{\text{p}}_{\text{s}}\rm {=}{\text{q}}^{\left| {\text{F}}^{\text{s}}_{\text{f}}\right| }{\rm {(1-q)}}^{\left| {\text{F}}^{\text{s}}_{\text{r}}\right| }\) .

In all experiments, we used \(\left| \mathcal {D}\right| \rm {=}{\text{F}}_{\text{U}}\bigcup {{\text{F}}_{\text{R}}}\rm {=12+8=20}\) sites which is a large-sized CRFLP problem and is more difficult to solve than the uncapacitated version (URFLP). The size of the failure scenario set is \(\ \left| \text{S}\right| \rm {=4,096}\) . The deterministic equivalent formulation has \(\text{20}\) binary \({\text{x}}_{\text{i}}\) \(\rm {1,720,320=}\left| \text{F}\right| {\times }\left| \text{D}\right| {\times }\left| \text{S} \right|\) \(=21\times 20\times 4,096\) continuous \({\text{y}}^{\text{s}}_{\text{ij}}\) variables. Further, there are \(\rm {1,888,256=}\) \(\rm {81,920+1,720,320+86,016=}\left| \text{D}\right| {\times }\left| \text{S}\right| \rm {+}\left| \text{F}\right| {\times }\left| \text{D}\right| {\times }\left| \text{S}\right| \rm {+\ }\left| \text{F}\right| {\times }\left| \text{S}\right|\) constraints corresponding to constraints (12)-(14) in the CRFLP formulation. Hence, the size of the constraint matrix of the deterministic equivalent MIP formulation is \({1,720,320\times 1,888,256}\) which cannot be tackled with exact solution procedures (e.g., branch-and-cut or column generation methods). Note that while solving LPs with this size is computationally feasible, the presence of the binary variables makes the solution a daunting task. We generated sample sets for SAA and the SBPHA (and d-SBPHA) by randomly sampling from \(\text{U}\left[ \text{0,1}\right]\) as follows. Given the scenario probabilities, \({\text{p}}_{\text{s}},\) we calculate the scenario cumulative probability vector \(\{{\text{p}}_{\text{1}},\left( {\text{p}}_{\text{1}}\rm {+}{\text{p}}_{\text{2}}\right) \rm {,\dots ,(\ }{\text{p}}_{\text{1}}\rm {+}{\text{p}}_{\text{2}}\rm {+\dots +}{\text{p}}_{\left| \text{S}\right| \rm {-1}}\rm {),\ 1}{\}}\) which has \(\rm {|S|}\) intervals. We first generate the random number and then select the scenario corresponding to the interval containing the random number. We tested the SAA, SBPHA, and d-SBPHA algorithms with varying number of samples \({\ (M)}\) , and sample sizes \({\ (N)}\) . Whenever possible, we use the same sample sets for all three methods. We select the reference set ( \({\text{N}}^{\rm {'}}\) ) as the entire scenario set, i.e., \({\text{N}}^{\rm {'}}\rm {=S}\) which is used to evaluate the second stage performance of a solution. We note that this is computationally tractable due to relatively small number of scenarios and that the second stage problem is an LP. In case of large scenario set or integer second stage problem, one should select \(\ {\text{N}}^{\rm {'}}{\ll }\text{S}\) .

4.3 Parameter sensitivity

In this subsection, we analyze the sensitivity of SBPHA with respect to the weight for the best incumbent solution parameter \(\rm {(}{\alpha }\rm {)}\) , penalty factor \(\rm {(}{\rho }\) ), and update parameter for the penalty factor \({\ (}{\beta }\rm {)}\) . Recall that \({\ }{\alpha }\) determines the bias of the best incumbent solution in obtaining the samples’ balanced solution, which is obtained as a weighted average of the best incumbent solution and the samples’ probability weighted solution. The parameter \({\rho }\) penalizes the Euclidean distance of a solution from the samples’ balanced solution and \({\beta }\) is the multiplicative update parameter for \({\rho }\) between two iterations. In all these tests, we set \(\ (\rm {M,\ N)=(5,\ 10)}\) , and \({\ q=0.3}\) unless otherwise is stated. We experimented with two \({\alpha }\) strategies, static and dynamic \(\ {\alpha }\) . We solved in total 480 \({(=10\ replications\ \times 48\ parameter\ settings)}\) problem instances.

The summary results of solving CRFLP using 10 independent sample sets (replications) with static strategy \({\alpha }\) =0.6, \(\ {\beta }\rm {=}{\{}\rm {1.1,1.2,1.3,1.4,1.5,1.8}{\}}\) , and \({\rho }\rm {=}{\{}\text{1,20,40,80,200}{\}}\) are presented in Table  1 . The detailed results of the 10 replications of Table  1 together with the detailed replication results with static strategy for \({\alpha }\) = \({\{}\) 0.6,0.7,0.8 \({\}}\) and dynamic strategy \({\Delta }{\alpha }\rm {=}{\{}\rm {0.02,0.03,0.05}{\}}\) are presented in Appendix A, Table  5 .

The first column in Table  1 shows the \({\alpha }\) strategy and its parameter value. Note that in the dynamic strategy, we select the initial value as \({\ }{{\alpha }}^{\rm {k=0}}\rm {=1}\) in Appendix A, Table  5 . The second and third columns show penalty factor \({\ (}{\rho }\) ) and update parameter for the penalty factor \({\ (}{\beta }\rm {)}\) , consecutively. The objective function values for the \(\text{10}\) replications (each replication consists of \({M=5}\) samples) are reported in columns \({\ 4-13}\) (shown only for replications \({\ 1}\) , \({\ 2}\) and \(\text{10}\) in Table  1 and detailed results are shown in Appendix A, Table  5 ). The first column under the “Objective” heading presents the average objective function value across \(\text{10}\) replications, and the second column under the “Objective” heading presents the optimality gap (i.e., \(\text{ga}{\text{p}}_{\text{1}}\) ) between the average replication solution and the best objective function value found, which is \({\ 8995.08}\) Footnote 1 , while the third and fourth columns under the “Objective” heading present the minimum and maximum objective values across \(\text{10}\) replications. Average objective function value and \(\text{ga}{\text{p}}_{\text{1}}\) are calculated as follows:

where \(\text{Rep}\) is the number of replications, e.g., \(\rm {Rep=10}\) in this section’s experiments.

In the last column, we report on the computational (CPU) time in seconds for tests. The complete results on CPU times are provided in Appendix A, Table  6 . First observation from Table  1 is that the SBPHA is relatively insensitive to the \(\alpha\) strategy employed and the parameter settings selected. Secondly, we observe that the performance of SBPHA with different parameter settings depends highly on the sample (see Table  5 ). As seen in Table  5 replication 7, most of the configurations show good performance as they all obtain the optimal solution. Further, as the \(\Delta _{ \alpha }\) increases, the best incumbent solution becomes increasingly more important leading to decreased computational time. While some parameter settings exhibit good performance in solution quality, their computational times are also higher, and vice versa.

In selecting the parameter settings for SBPHA, we are interested in a parameter setting that offers a balanced trade-off between the solution quality and the solution time. In order to determine such parameter settings, we developed a simple, yet effective, parameter configuration selection index. The selection criterion is the product of average \(gap_{1}\) and CPU time. Clearly, smaller the index value, the better is the performance of the corresponding parameter configuration. To illustrate, using the results of the first row in Table  1 , the index of static \(\alpha =0.6\) , with starting penalty factor \(\left( \rho \right)\) \(=1\) and penalty factor update \(\beta =1.8\) , is calculated as \(19.00 \left( =4.039\% \times 470.4 \right)\) . Parameter selection indexes corresponding to all 480 experiments are shown in Table  2 . According to the aggregate results in Table  2 (the ‘Total’ row at the bottom), the static \(\alpha =0.7\) setting is the best, the static \(\alpha =0.6\) is the second best, and dynamic \(\alpha\) with \(\Delta _{ \alpha }=0.03\) provides the third best performance. Hence, we use only these \(\alpha\) parameter configurations in our experiments. In terms of penalty parameter configuration, we select the best setting, i.e., starting penalty factor \(\left( \rho \right) =200\) and penalty factor update \(\beta =1.1\) for all \(\alpha\) parameter configurations. Note that by selecting a larger starting penalty factor, the SBPHA would converge faster but the quality of the solution converged would be lower. Therefore, we restricted our experiments in terms of penalty factor to be at most 200 ( \(\rho\) ).

Further, among all the experiments, the best parameter configuration in terms of index is with static \(\alpha =0.6\) , when starting penalty factor \(\rho =80\) and update parameter \(\left( \beta \right) =1.2\) . In Tables  1 and   5 , this configuration of parameter selection provides the best average gap performance and a good CPU time performance. Hence we also included this parameter configuration in our computational performance experiments.

In the remainder of the computational experiments we used sample size and number as \(\left( M. N \right) = \left( 5. 10 \right)\) . which enables the SBPHA to search the solution space while maintaining computational time efficiency.

4.4 Computational performance of SBPHA and d-SBPHA

In this sub-section, we first show the performance of the d-SBPHA on improving the solution quality of SBPHA and then compare the performances of the SAA and the proposed d -SBPHA algorithm. In the remainder of the experiments, with an abuse of the optimality definition, we refer to the best solution as the “exact solution" . This solution is obtained by selecting the best amongst all SBPHA, d -SBPHA, and SAA solutions and the time-restricted solution of the CPLEX.

4.4.1 Analysis of d-SBPHA and SBPHA

Figure  1 shows effect of discarding strategy on the solution quality for different facility failure probabilities. In all figures, results are based on the average of 10 replications. Optimality gap (shown as ‘Gap’) is calculated as in (18) but substituted \(v_{r}^{SBPHA}\) with \(v_{best.r}^{d-SBPHA}\) in (17) to calculate \(v_{Rep}^{Average}\) . First observation is that the d-SBPHA not only improves solution quality but also finds the optimal solution in most facility failure probabilities cases. When failure probability is \(q=0.1\) , d-SBPHA converges to optimal solution in less than 5 discarding iterations with all parameter configurations (Fig.  1 a). When \(q=0.2\) , d-SBPHA converges to optimal solution in all static \(\alpha\) strategies in less than 5 discarding strategies, and less than \(0.2\%\) optimality gap with dynamic \(\alpha\) strategy (Fig.  1 b).

Effect of discarding strategy on the solution quality for CRFLP with facility failure probabilities a \(q=0.1\) , b \(q=0.2\) , c \(q=0.3\) and d when q is random

Further, when failure probability is 0.3, d-SBPHA is not able converge to optimal solution; however it converges to solutions that are less than \(1\%\) away from the optimal on average (Fig.  1 c). Note that these results are based on the average of 10 replications, and at least 5 out of 10 replications are converged to optimal solution in all parameter configurations. Detailed results are provided in the next section.

Lastly, when failure probability is randomized, d-SBPHA converges to optimal solution in 10 or less discarding iterations, with three out of the four selected parameter configurations and less than \(0.5\%\) gap (Fig.  1 d). Hence, we conclude that discarding improves the solution performance and the improvement rates depend on the parameter configuration and the problem parameters. Reader is referred to Table  8 in Appendix A for detailed results.

Next, we present the CPU time performance of d-SBPHA for 10 discarding iterations (Fig.  2 ). Note that time plotted is cumulative over discarding iterations. Time of each discarding iteration is based on average of 10 replications and is reported in seconds.

First observation is that the CPU time is linearly increasing or increasing at a decreasing rate. Further, the solution time is similar for all facility failure probabilities (Fig.  2 a–d) with all parameter configurations.

CPU time performance of discarding strategy for CRFLP with facility failure probabilities a \(q=0.1\) , b \(q=0.2\) , c \(q=0.3\) , and d when q is random

All results are provided in the Appendix A in Table  9 . One main reason for linearly increasing CPU time is that the d-SBPHA makes use of the previously encountered solution information. In particular, dSBPHA does not test the reference sample ( \(N'\) ) performance of any first stage solution that is encountered and already tested before. This is achieved by maintaining a library of first stage solutions and corresponding reference sample ( \(N'\) ) performance in a dynamic table.

4.4.2 SAA, SBPHA and d-SBPHA comparison

In this sub-section, we compare the performances of the SBPHA, d-SBPHA, and SAA. First, we analyze the performance of the proposed SBPHA and d-SBPHA with respect to that of the exact method and the SAA method with different sample sizes (N) and number of samples (M). Here, for the sake of simplicity and to explain clearer, we randomly selected a parameter configuration, in which \(\alpha =0.7. \rho =200\) , and \(\beta =1.1\) , which is not the worst or the best parameter configuration.

Tables  3 and   4 illustrate these benchmark results for \(q= \{ 0.1, 0.2, 0.3, \text {and} \ \text {random} \}\) for one of the replications and, then, the average results across all replications are shown in Fig.  3 . The second column for SAA shows number of samples and sample size. i.e., ( M ,  N ). For dSBPHA, it shows number of replications \(r, \left( M,N \right)\) , and number of discarding iterations ( d ). Note that when the number of discarding iterations \(d=0\) , dSBPHA becomes SBPHA. Third column, \(F^{*},\) indicates the solution converged by each method, e.g., facilities opened. For instance with \(q=0.1 \ \text {and} \ \left( M, N \right) = \left( 5, 10 \right)\) , the SAA’s solution is to open facilities \(F^{*}= \{ 1, 2, 8, 10, 12 \}\) whereas the SBPHA opens facilities \(F^{*}= \{ 1, 2, 4, 10, 12 \}\) . 2-SBPHA and exact (optimal) solution opens facilities \(F^{*}= \{ 1, 2, 10, 11, 12 \}\) .

Fourth column presents the objective function value for SAA, SBPHA, d-SBPHA and exact method, \(v^{SAA}, v^{SBPHA}, v_{best}^{d-SBPHA}\) and \(v^{*}\) . Fifth column presents CPU time and the sixth column shows the optimality gap \(\left( Gap_{2} \right)\) measures. Reported time for d-SBPHA is the average time of converged solution that is found during the discarding iterations. Gap 2 for SAA, SBPHA and d -SBPHA uses the optimal solution value \(v^{*},\) and it is defined as.

Tables  3 and   4 show that with larger sample sizes the objective function on the SAA’s objective function is not always monotonously decreasing while the CPU time increases exponentially. The observation about the time is in accordance with those in Fig.  3 . SAA finds optimal solution only when \(N=75\) for \(q=0.1\) and cannot find optimal solution for \(q=0.2\) with any of M-N configurations. SAA also finds optimal solution for \(q=0.3\) only when M =20 and N =75 in more than 13, 000 seconds and shows relatively good performance for random q when M= 20. d-SBPHA finds optimal solution in all facility failure probabilities.

Effect of sample size on the solution quality and CPU time performance of SAA in comparison with d-SBPHA for CRFLP with facility failure probabilities a \(q=0.1\) , b \(q=0.2\) , c \(q=0.3\) , and d q random

Results for dSBPHA ( \(d=10\) ) in Fig.  3 are for all four parameter settings; first setting is for static \(\alpha =0.6, \ \rho =200\) and \(\beta =1.1\) , second is static \(\alpha =0.7, \ \rho =200\) and \(\beta =1.1,\) third is static \(\alpha =0.6, \ \rho =80\) and \(\beta =1.2\) and fourth one is dynamic \(\Delta _{ \alpha }=0.03, \ \rho =200\) and \(\beta =1.1\) .

In Fig.  3 , we present the CPU time and solution quality performances of the SAA for N= \(\{ 10, 25, 40, 50, 75 \}\) sample sizes and compare with that of the proposed d-SBPHA, in which \(d=10\) and N=10 in solving CRFLP with failure probabilities \(q= \{ 0.1, 0.2, 0.3, random \}\) . We use 5 (M) samples in both SAA and four different parameter configurations in the proposed method. Different number of samples would increase the solution time of SAA and SBPHA. However, as seen from results, SBPHA already captures the optimal solution in majority of the experiments with only with five samples in less time than the SAA. Therefore, increasing number of samples would not make any difference on the comparison of the SAA and SBPHA in terms of performance. The results indicate that the sample size effect the SAA’s CPU time. For instance, the CPU time of the SAA is growing exponentially. Furthermore, in none of the failure probability cases, the solution quality performance of SAA has converged to that of d-SBPHA. The solution quality of all four configurations of the proposed d-SBPHA are either optimal or near optimal. The gap in Fig.  3 is calculated as in Fig.  1 and the CPU time of d-SBPHA shows the average CPU time when the converged solution is found during the discarding iterations.

5 Conclusion

In this study we have developed hybrid algorithms for two-stage SP problems, specifically for CRFLP. The proposed algorithms are hybridization of two existing methods. The first one is the Monte Carlo sampling based algorithm, which is called SAA. SAA method provides an attractive approximation for SP problems when the number of uncertain parameters increases. The second algorithm is PHA, which is an exact solution methodology for SP problems. The research presented in this study mainly addresses two issues that arise when using SAA and PHA methods individually; lack of effectiveness in solution quality of SAA and lack of efficiency in computational time of PHA.

The first proposed algorithm is called SBPHA, which is the integration of SAA and PHA. This integration considers each sample as a small deterministic problem and employs the SAA algorithm iteratively. In each iteration, the non-anticipativity constraints is injected into the solution process by introducing penalty terms in the objective that guides the solution of each sample to the samples’ balanced solution and to ensure that non-anticipativity constraints are satisfied. The two key parameters of SBPHA are the weight of the best incumbent solution and the penalty factor.

The weight for the best incumbent solution adjusts the importance given to samples’ best found solution and to the most recent average sample solution in calculating the balanced solution. The penalty factor modulates the rate at which the sample solutions converge to the samples’ balanced solution. Given that the best found solution improves over time, we propose two strategies for the weight of the best incumbent solution: static versus dynamic strategy.

We first conducted experiments for sensitivity analysis of the algorithm with respect to the parameters. The results show that the SBPHA’s solution quality performance is relatively insensitive to the choice of strategy for the weight of the best incumbent solution. i.e., both the static and dynamic strategies are able to converge to the optimum solution. SBPHA is able to converge to the optimal solution even with small number of samples and small sample sizes.

In addition to the sensitivity experiments, we compared the performances of SBPHA and d-SBPHA with SAA’s. These results show that the SBPHA and d-SBPHA are able to improve the solution quality noticeably with reasonable computational effort compared to SAA. Further, increasing SAA’s sample size to match the solution quality performance of SBPHA or dSBPHA requires significant computational effort which is not affordable in many practical instances.

The contributions of this research are as follows:

Contribution 1: Developed SBPHA, which provides a configurable solution method that improves the sampling based methods’ accuracy and PHA’s efficiency for stochastic two-stage CRFLP.

Contribution 2: Enhanced SBPHA for SPs with binary first stage decision variables. The improved algorithm is called Discarding-SBPHA (d-SBPHA). It is analytically proved that SBPHA guarantee optimal solution to the mentioned problems when the number of discarding iterations approaches to infinity.

In addition to all the advantages introduced by the proposed algorithms, there are some limitations to discuss. First, the proposed algorithms provide very accurate solutions to the problems that include binary first stage decision variables. However, the cases with general integer or continuous first stage decision variables are not tested and the performances of the proposed algorithms need to be examined. Second, the experimental study conducted on a problem that has continuous second stage decision variables. Even though the SBPHA does not require continuous second stage decision variables, this circumstance still need to be inspected. Lastly, the hyper-tuning parameters of the algorithms need to be re-optimized when the application area or the dynamics of the problem changes.

Besides and parallel to the limitations mentioned, there are several possible avenues for future research on the algorithms. First opportunity is to investigate the integration of alternative solution methodologies to improve the convergence rate and solution quality, i.e., Stochastic Decomposition (SD), Stochastic Dual Dynamic Programming (SDDP), L-Shaped decomposition, Benders Decomposition (BD) etc. Second extension is the research on the application of the d-SBPHA to the SPs that have linear first stage decision variables. Third extension is the investigations on the development of a general strategy for the SBPHA and d-SBPHA specific parameters to reduce the computational effort spent on the parameter sensitivity steps. Fourth extension is the adaptation of the ideas of d-SBPHA to multistage SPs. Lastly, using proposed algorithms a black box type optimization tool may be developed for the two-stage SP problems, i.e., finance (Uğur et al. 2017 ; Savku and Weber 2018 ).

The best solution is obtained by selecting the best amongst all SBPHA solutions (e.g., out of 480 solutions) and the time-restricted solution of the CPLEX. The latter solution is obtained by solving the deterministic equivalent using CPLEX method with %0.05 optimality gap tolerance and 10 hours (36,000 seconds) of time limit until either the CPU time-limit is exceeded or the CPLEX terminates due to insufficient memory.

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Acknowledgements

Research of the third author was partly supported by the US National Science Foundation under Grants DMS-1512846 and DMS-1808978, by the US Air Force Office of Scientific Research under Grant #15RT0462, and by the Australian Research Council Discovery Project DP-190100555.

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Department of Industrial Engineering, Yildiz Technical University, Besiktas/Istanbul, 34349, Turkey

Nezir Aydin

College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar

Department of Industrial and Systems Engineering, Wayne State University, Detroit, MI, 48202, USA

Alper Murat

Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

Boris S. Mordukhovich

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Contributions

N.A., A.M. and B.M. did the conceptualization, N.A. and A.M.wrote the main manuscript text, N.A. prepared figures and tables, all authors reviewed the manuscript.

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Correspondence to Nezir Aydin .

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Supplemental results for SBPHA and dSBPHA

See Tables 5 , 6 , 7 , 8 , 9 .

Data set used for CRFLP

First column of Table  10 shows whether the possible facility site is reliable or not. Second column shows the facility number. Third and fourth columns show the location (lat-long) of the facility sites. Fifth column presents the demand of the location and sixth column presents the fix opening cost that is going to be applied if a facility is opened at the specified location. Lastly the seventh column presents the failure probability if it is a random failure cases, otherwise all values in this column (only rows 1-12) are equal, e.g.. \(q=q_{i}=0.1\) . Emergency cost (demand satisfying cost if the demand is not satisfied from an opened facility but from emergency. e.g., dummy facility) is 20 and is equal for all facility sites. Capacity for all facilities is taken 2000.

Flowcharts of SBPHA and d-SBPHA

See Figs. 4 and 5 .

Flow chart of SBPHA

Flow chart of d-SBPHA

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Aydin, N., Murat, A. & Mordukhovich, B.S. Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem. Artif Intell Rev 57 , 135 (2024). https://doi.org/10.1007/s10462-024-10755-w

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Accepted : 26 March 2024

Published : 07 May 2024

DOI : https://doi.org/10.1007/s10462-024-10755-w

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