engineering problem solving method

Oct 13, 2019

10 Steps to Problem Solving for Engineers

Updated: Dec 6, 2020

With the official launch of the engineering book 10+1 Steps to Problem Solving: An Engineer's Guide it may be interesting to know that formalization of the concept began in episode 2 of the Engineering IRL Podcast back in July 2018.

As noted in the book remnants of the steps had existed throughout my career and in this episode I actually recorded the episode off the top of my head.

My goal was to help engineers build a practical approach to problem solving.

Have a listen.

Who can advise on the best approach to problem solving other than the professional problem solvers - Yes. I'm talking about being an Engineer.

There are 2 main trains of thought with Engineering work for non-engineers and that's trying to change the world with leading edge tech and innovations, or plain old boring math nerd type things.

Whilst, somewhat the case what this means is most content I read around Tech and Engineering are either super technical and (excruciatingly) detailed. OR really riff raff at the high level reveling at the possibilities of changing the world as we know it. And so what we end up with is a base (engineer only details) and the topping (media innovation coverage) but what about the meat? The contents?

There's a lot of beauty and interesting things there too. And what's the centrepiece? The common ground between all engineers? Problem solving.

The number one thing an Engineer does is problem solving. Now you may say, "hey, that's the same as my profession" - well this would be true for virtually every single profession on earth. This is not saying there isn't problem solving required in other professions. Some problems require very basic problem solving techniques such is used in every day life, but sometimes problems get more complicated, maybe they involve other parties, maybe its a specific quirk of the system in a specific scenario. One thing you learn in engineering is that not all problems are equal. These are

The stages of problem solving like a pro:

Is the problem identified (no, really, are you actually asking the right question?)

Have you applied related troubleshooting step to above problem?

Have you applied basic troubleshooting steps (i.e. check if its plugged in, turned it on and off again, checked your basics)

Tried step 2 again? (Desperation seeps in, but check your bases)

Asked a colleague or someone else that may have dealt with your problem? (50/50 at this point)

Asked DR. Google (This is still ok)

Deployed RTFM protocol (Read the F***ing Manual - Engineers are notorious for not doing this)

Repeated tests, changing slight things, checking relation to time, or number of people, or location or environment (we are getting DEEP now)

Go to the bottom level, in networking this is packet sniffers to inspect packets, in systems this is taking systems apart and testing in isolation, in software this is checking if 1 equals 1, you are trying to prove basic human facts that everyone knows. If 1 is not equal to 1, you're in deep trouble.At this point you are at rebuild from scratch, re install, start again as your answer (extremely expensive, very rare)

And there you have it! Those are your levels of problem solving. As you go through each step, the more expensive the problem is. -- BUT WAIT. I picked something up along the way and this is where I typically thrive. Somewhere between problem solving step 8 and 10.

The secret step

My recommendation at this point is to try tests that are seemingly unrelated to anything to do with the problem at all.Pull a random cable, test with a random system off/on, try it at a specific time of the day, try it specifically after restarting or replugging something in. Now, not completely random but within some sort of scope. These test are the ones that when someone is having a problem when you suggest they say "that shouldn't fix the problem, that shouldn't be related" and they are absolutely correct.But here's the thing -- at this stage they have already tried everything that SHOULD fix the problem. Now it's time for the hail mary's, the long shots, the clutching at straws. This method works wonders for many reasons. 1. You really are trying to try "anything" at this point.

2. Most of the time we may think we have problem solving step number 1 covered, but we really don't.

3. Triggering correlations.

This is important.

Triggering correlations

In a later post I will cover correlation vs causation, but for now understand that sometimes all you want to do is throw in new inputs to the system or problem you are solving in order to get clues or re identify problems or give new ways to approach earlier problem solving steps. There you have it. Problem solve like a ninja. Approach that extremely experienced and smart person what their problem and as they describe all the things they've tried, throw in a random thing they haven't tried. And when they say, well that shouldn't fix it, you ask them, well if you've exhausted everything that should have worked, this is the time to try things that shouldn't. Either they will think of more tests they haven't considered so as to avoid doing your preposterous idea OR they try it and get a new clue to their problem. Heck, at worst they confirm that they do know SOMETHING about the system.

Go out and problem solve ! As always, thanks for reading and good luck with all of your side hustles.

If you prefer to listen to learn we got you covered with the Engineering IRL show!

For Youtube please go to:

https://youtu.be/EHaRNZhqmHA

For Spotify please go to:

https://open.spotify.com/show/3UZPfOvNwQkaCA1jLIOxp4

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Technical Tactics
10+1 Steps to Problem Solving

Engineering Problem Solving ¶

Some problems are so complex that you have to be highly intelligent and well-informed just to be undecided about them. —Laurence J. Peter

Steps in solving ‘real world’ engineering problems ¶

The following are the steps as enumerated in your textbook:

Collaboratively define the problem

List possible solutions

Evaluate and rank the possible solutions

Develop a detailed plan for the most attractive solution(s)

Re-evaluate the plan to check desirability

Implement the plan

Check the results

A critical part of the analysis process is the ‘last’ step: checking and verifying the results.

Depending on the circumstances, errors in an analysis, procedure, or implementation can have significant, adverse consequences (NASA Mars orbiter crash, Bhopal chemical leak tragedy, Hubble telescope vision issue, Y2K fiasco, BP oil rig blowout, …).

In a practical sense, these checks must be part of a comprehensive risk management strategy.

My experience with problem solving in industry was pretty close to this, though encumbered by numerous business practices (e.g., ‘go/no-go’ tollgates, complex approval processes and procedures).

In addition, solving problems in the ‘real world’ requires a multidisciplinary effort, involving people with various expertise: engineering, manufacturing, supply chain, legal, marketing, product service and warranty, …

Exercise: Problem solving

Step 3 above refers to ranking of alternatives.

Think of an existing product of interest.

What do you think was ranked highest when the product was developed?

Consider what would have happened if a different ranking was used. What would have changed about the product?

Brainstorm ideas with the students around you.

Defining problems collaboratively ¶

Especially in light of global engineering , we need to consider different perspectives as we define our problem. Let’s break the procedure down into steps:

Identify each perspective that is involved in the decision you face. Remember that problems often mean different things in different perspectives. Relevant differences might include national expectations, organizational positions, disciplines, career trajectories, etc. Consider using the mnemonic device “Location, Knowledge, and Desire.”

Location : Who is defining the problem? Where are they located or how are they positioned? How do they get in their positions? Do you know anything about the history of their positions, and what led to the particular configuration of positions you have today on the job? Where are the key boundaries among different types of groups, and where are the alliances?

Knowledge : What forms of knowledge do the representatives of each perspective have? How do they understand the problem at hand? What are their assumptions? From what sources did they gain their knowledge? How did their knowledge evolve?

Desire : What do the proponents of each perspective want? What are their objectives? How do these desires develop? Where are they trying to go? Learn what you can about the history of the issue at hand. Who might have gained or lost ground in previous encounters? How does each perspective view itself at present in relation to those it envisions as relevant to its future?

As formal problem definitions emerge, ask “Whose definition is this?” Remember that “defining the problem clearly” may very well assert one perspective at the expense of others. Once we think about problem solving in relation to people, we can begin to see that the very act of drawing a boundary around a problem has non-technical, or political dimensions, depending on who controls the definition, because someone gains a little power and someone loses a little power.

Map what alternative problem definitions mean to different participants. More than likely you will best understand problem definitions that fit your perspective. But ask “Does it fit other perspectives as well?” Look at those who hold Perspective A. Does your definition fit their location, their knowledge, and their desires? Now turn to those who hold Perspective B. Does your definition fit their location, knowledge, and desires? Completing this step is difficult because it requires stepping outside of one’s own perspective and attempting to understand the problem in terms of different perspectives.

To the extent you encounter disagreement or conclude that the achievement of it is insufficient, begin asking yourself the following: How might I adapt my problem definition to take account of other perspectives out there? Is there some way of accommodating myself to other perspectives rather than just demanding that the others simply recognize the inherent value and rationality of mine? Is there room for compromise among contrasting perspectives?

How ‘good’ a solution do you need ¶

There is also an important aspect of real-world problem solving that is rarely articulated and that is the idea that the ‘quality’ of the analysis and the resources expended should be dependent on the context.

This is difficult to assess without some experience in the particular environment.

How ‘Good’ a Solution Do You Need?

Some rough examples:

10 second answer (answering a question at a meeting in front of your manager or vice president)

10 minute answer (answering a quick question from a colleague)

10 hour answer (answering a request from an important customer)

10 day answer (assembling information as part of a trouble-shooting team)

10 month answer (putting together a comprehensive portfolio of information as part of the design for a new $200,000,000 chemical plant)

Engineering Method

The engineering method (also known as engineering design) is a systematic approach used to reach the desired solution to a problem. There are six steps (or phases): idea, concept, planning, design, development, and launch from problem definition to desired result.

Engineering Method. Source: Ronald L. Lasser

The engineering method has six steps (or phases):

Development

The development step is often divided to include the iterative cycle of build, test, debug, and redesign. The engineering method by nature is an iterative process.

The idea phase usually begins with a problem. The problem statement is typically only vaguely defined and requires research into its viability and its feasibility. Viability suggests that there is significant value (or demand in the case of product development) in pursing the solution. Feasibility serves as a check on whether the idea can be realized. Feasibility may be high, medium, or low: where high feasibility means that people, technology, and time resources are readily available or known; medium is that resources may not be available directly, but can be found; and low means the resources may be rare or do not exist. The most critical part of the idea phase is to define the problem, validate its value, and identify the customer who desires its solution.

The concept phase is about generating numerous models (mathematical, physical, simulation, simple drawings or sketches), all of which should convey that the solution meets the customer’s expectations or requirements. The numerous concepts are generated using brainstorming techniques, which are review sessions in which elements of one concept are recombined with elements from other in an effort to find a single concept that fits best. Typical design judgment and compromise are required to merge concepts. The concept phase ends with a selection of a single concept.

3. Planning

The planning phase is about defining the implementation plan: identifying the people, tasks, task durations, task dependencies, task interconnections, and budget required to get the project done. Many tools are used to convey this information to team members and other stakeholders including Gantt and Pert charts, resource loading spreadsheets, sketches, drawings, proof-of-concept models to validate that the project can be successfully completed.

One critical tool of the planning phase is the system engineering diagram. This diagram shows the solution as an interconnection of smaller and less complicated sub-systems. A system engineering diagram establishes all the inputs and outputs for each module, as well as the way in which the module transforms the inputs into outputs.

The design phase is where “the rubber meets the road.” Details are specified; specifications are established. Some call this phase “design planning” and the development phase “detailed design.” But no matter what it is called, the purpose of this phase is to translate the customer requirements and systems engineering model into engineering specifications that an engineer (designer) can work with to design and build a working prototype. Specifications are detailed using a number with associated units, e.g., 4 volts, or 3.82 inches, or 58 Hz, or a completion time of 22 days.

5. Development

The purpose of development is to generate the engineering documentation: schematics, drawings, source code, and other design information into a working prototype that demonstrates the solution to the problem. The solution may be a tangible working prototype or an intangible working simulation. Of course, nothing works the first time, so this part of the process tends to be more iterative than the other phases. Specifically, it consists of the iterative cycle: design, test, debug, and redesign. If the project had earlier delays or is not on the planned schedule for other reasons, then this time may be the most frantic since the customer deadline may be closely looming.

While testing and debug are often consider a separate phase, most times they occur side-by-side with development as a design morphs from a concept to an artifact. The latter is recommended, reserving time at the end of development for a final test to confirm the desired result meets customer expectation and designer’s intent. Testing is the verification and validation phase where the concept meets both the anticipated design specifications and the customer’s requirements of the solution. Testing is achieved through experiments—an information-gathering method where dissimilarity and difference are assessed with respect to the design’s present and compared to desired state for the design. The purpose of an experiment is to determine whether test results agree or conflict with the a priori stated behavior. A sufficient numbers of successful testing verifications and validations are necessary to generate acceptable results and to reduce any risk that the desired behavior is present and functions as expected. If the test observations and results do not agree, then a debug process is necessary to identify the root causes and begin corrective action to resolve the discrepancies.

Launch includes the release of the engineering design and documentation package to manufacturing facilities for production. At this point, all qualification testing is complete, and the working prototype has demonstrated functionality.

Cited References

Ertas, A., & Jones, J. C. (1996). The Engineering design process (2nd ed.). New York: John Wiley & Sons. OCLC WorldCat Permalink: http://www.worldcat.org/oclc/807148675
Ullman, D. G. (2009). The Mechanical Design Process (4th ed.). New York, N.Y.: McGraw Hill. OCLC WorldCat Permalink: http://www.worldcat.org/oclc/244060468
Ulrich, K.T., & Eppinger, S. D. (2008). Product Design and Development (4th ed.) New York, N.Y.: McGraw Hill. OCLC WorldCat Permalink: http://www.worldcat.org/oclc/122424997
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Solving Everyday Problems Using the Engineering Design Cycle

Hands-on Activity Solving Everyday Problems Using the Engineering Design Cycle

Grade Level: 7 (6-8)

(two 60-minutes class periods)

Additional materials are required if the optional design/build activity extension is conducted.

Group Size: 4

Activity Dependency: None

Subject Areas: Science and Technology

NGSS Performance Expectations:

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This activity introduces students to the steps of the engineering design process. Engineers use the engineering design process when brainstorming solutions to real-life problems; they develop these solutions by testing and redesigning prototypes that work within given constraints. For example, biomedical engineers who design new pacemakers are challenged to create devices that help to control the heart while being small enough to enable patients to move around in their daily lives.

After this activity, students should be able to:

Explain the stages/steps of the engineering design process .
Identify the engineering design process steps in a case study of a design/build example solution.
Determine whether a design solution meets the project criteria and constraints.
Think of daily life situations/problems that could be improved.
Apply the engineering design process steps to develop their own innovations to real-life problems.
Apply the engineering design cycle steps to future engineering assignments.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science, international technology and engineering educators association - technology.

View aligned curriculum

Do you agree with this alignment? Thanks for your feedback!

State Standards

Massachusetts - science.

Each group needs:

Marisol Case Study , one per student
Group Leader Discussion Sheet , one per group

To share with the entire class:

computer/projector setup to show the class the Introduction to the Engineering Design Cycle Presentation , a Microsoft® PowerPoint® file
paper and pencils
(optional) an assortment of scrap materials such as fabric, super glue, wood, paper, plastic, etc., provided by the teacher and/or contributed by students, to conduct the hands-on design/build extension activity

(Have the 19-slide Introduction to the Engineering Design Cycle Presentation , a PowerPoint® file, ready to show the class.)

Have you ever experienced a problem and wanted a solution to it? Maybe it was a broken backpack strap, a bookshelf that just kept falling over, or stuff spilling out of your closet? (Let students share some simple problems with the class). With a little bit of creativity and a good understanding of the engineering design process, you can find the solutions to many of these problems yourself!

But what is the engineering design process? (Listen to some student ideas shared with the class.) The engineering design process, or cycle, is a series of steps used by engineers to guide them as they solve problems.

(Show students the slide presentation. Refer to the notes under each slide for a suggested script and comments. The slides introduce the main steps of the engineering design process, and walk through a classroom problem—a teacher’s disorganized desk that is preventing timely return of graded papers—and how students devise a solution. It also describes the work of famous people—Katherine Johnson, Lee Anne Walters, Marc Edwards, James E. West and Jorge Odón—to illustrate successful examples of using the steps of the engineering design process.)

Now that we’ve explore the engineering design process, let’s see if we can solve a real-world problem. Marisol is a high-school student who is very excited to have their own locker. They have lots of books, assignments, papers and other items that they keep in their locker. However, Marisol is not very organized. Sometimes they are late to class because they need extra time to find things that were stuffed into their locker. What is Marisol’s problem? (Answer: Their locker is disorganized.) In your groups, you’ll read through Marisol’s situation and see how they use the engineering design process to solve it. Let’s get started!

This activity is intended as an introduction to the engineering design cycle. It is meant to be relatable to students and serve as a jumping off point for future engineering design work.

Engineers follow the steps of the engineering design process to guide them as they solve problems. The steps shown in Figure 1 are:

Ask: identify the need & constraints

Identify and define the problem. Who does the problem affect? What needs to be accomplished? What is the overall goal of the project?
Identify the criteria and constraints. The criteria are the requirements the solution must meet, such as designing a bag to hold at least 10 lbs. Constraints are the limitations and restrictions on a solution, such as a maximum budget or specific dimensions.

Research the problem

Learn everything you can about the problem. Talk to experts and/or research what products or solutions already exist.
If working for a client, such as designing new filters for a drinking water treatment plant, talk with the client to determine the needs and wants.

Imagine: develop possible solutions

Brainstorm ideas and come up with as many solutions as possible. Wild and crazy ideas are welcome! Encourage teamwork and building on ideas.

Plan: select a promising solution

Consider the pros and cons of all possible solutions, keeping in mind the criteria and constraints.
Choose one solution and make a plan to move forward with it.

Create: build a prototype

Create your chosen solution! Push for creativity, imagination and excellence in the design.

Test and evaluate prototype

Test out the solution to see how well it works. Does it meet all the criteria and solve the need? Does it stay within the constraints? Talk about what worked during testing and what didn’t work. Communicate the results and get feedback. What could be improved?

Improve: redesign as needed

Optimize the solution. Redesign parts that didn’t work, and test again.
Iterate! Engineers improve their ideas and designs many times as they work towards a solution.

Some depictions of the engineering design process delineate a separate step—communication. In the Figure 1 graphic, communication is considered to be incorporated throughout the process. For this activity, we call out a final step— communicate the solution —as a concluding stage to explain to others how the solution was designed, why it is useful, and how they might benefit from it. See the diagram on slide 3.

For another introductory overview of engineering and design, see the What Is Engineering? What Is Design? lesson and/or show students the What Is Engineering? video.

Before the Activity

Make copies of the five-page Marisol Case Study , one per student, and the Group Leader Discussion Sheet , one per group.
Be ready to show the class the Introduction to the Engineering Design Cycle Presentation , a PowerPoint® file.

With the Students

As a pre-activity assessment, spend a few minutes asking students the questions provided in the Assessment section.
Present the Introduction/Motivation content to the class, which includes using the slide presentation to introduce students to the engineering design cycle. Throughout, ask for their feedback, for example, any criteria or constraints that they would add, other design ideas or modifications, and so forth.
Divide the class into groups of four. Ask each team to elect a group leader. Hand out the case study packets to each student. Provide each group leader with a discussion sheet.
In their groups, have students work through the case study together.
Alert students to the case study layout with its clearly labeled “stop” points, and direct them to just read section by section, not reading beyond those points.
Suggest that students either taking turns reading each section aloud or read each section silently.
Once all students in a group have read a section, the group leader refers to the discussion sheet and asks its questions of the group, facilitating a discussion that involves every student.
Encourage students to annotate the case study as they like; for example, they might note in the margins Marisol’s stage in the design process at various points.
As students work in their groups, walk around the classroom and encourage group discussion. Ensure that each group member contributes to the discussion and that group members are focused on the same section (no reading ahead).
After all teams have finished the case study and its discussion questions, facilitate a class discussion about how Marisol used the engineering design cycle. This might include referring back to questions 4 and 5 in “Stop 5” to discuss remaining questions about the case study and relate the case study example back to the community problems students suggested in the pre-activity assessment.
Administer the post-activity assessment.

brainstorming: A team creativity activity with the purpose to generate a large number of potential solutions to a design challenge.

constraint: A limitation or restriction. For engineers, design constraints are the requirements and limitations that the final design solutions must meet. Constraints are part of identifying and defining a problem, the first stage of the engineering design cycle.

criteria: For engineers, the specifications and requirements design solutions must meet. Criteria are part of identifying and defining a problem, the first stage of the engineering design cycle.

develop : In the engineering design cycle, to create different solutions to an engineering problem.

engineering: Creating new things for the benefit of humanity and our world. Designing and building products, structures, machines and systems that solve problems. The “E” in STEM.

engineering design process: A series of steps used by engineering teams to guide them as they develop new solutions, products or systems. The process is cyclical and iterative. Also called the engineering design cycle.

evaluate: To assess something (such as a design solution) and form an idea about its merit or value (such as whether it meets project criteria and constraints).

optimize: To make the solution better after testing. Part of the engineering design cycle.

Pre-Activity Assessment

Intro Discussion: To gauge how much students already know about the activity topic and start students thinking about potential design problems in their everyday lives, facilitate a brief class discussion by asking students the following questions:

What do engineers do? (Example possible answers: Engineers design things that help people, they design/build/create new things, they work on computers, they solve problems, they create things that have never existed before, etc.)
What are some problems in your home, school or community that could be solved through engineering? (Example possible answers: It is too dark in a community field/park at night, it is hard to carry shopping bags in grocery store carts, the dishwasher does not clean the dishes well, we spend too much time trying to find shoes—or other items—in the house/garage/classroom, etc.)
How do engineers solve problems? (Example possible answers: They build new things, design new things, etc. If not mentioned, introduce students to the idea of the engineering design cycle. Liken this to how research scientists are guided by the steps of the scientific method.)

Activity Embedded Assessment

Small Group Discussions: As students work, observe their group discussions. Make sure the group leaders go through all the questions for each section, and that each group member contributes to the discussions.

Post-Activity Assessment

Marisol’s Design Process: Provide students with writing paper and have them write “Marisol’s Design Process” at the top. Direct them to clearly write out the steps that Marisol went through as they designed and completed their locker organizer design and label them according to where they fit in the engineering design cycle. For example, “Marisol had to jump back to avoid objects falling out of their locker” and they stated a desire to “wanted to find a way to organize their locker” both illustrate the “identifying the problem” step.

Which part of the engineering design cycle is Marisol working on as they design an organizer?
Why is it important to identify the criteria and constraints of a project before building and testing a prototype? (Example possible answers: So that the prototype will be the right size, so that you do not go over budget, so that it will solve the problem, etc.)
Why do engineers improve and optimize their designs? (Example possible answers: To make it work better, to fix unexpected problems that come up during testing, etc.)

To make this a more hands-on activity, have students design and build their own locker organizers (or other solutions to real-life problems they identified) in tandem with the above-described activity, incorporating the following changes/additions to the process:

Before the activity: Inform students that they will be undertaking an engineering design challenge. Without handing out the case study packet, introduce students to Marisol’s problem: a disorganized locker. Ask students to bring materials from home that they think could help solve this problem. Then, gather assorted materials (wood and fabric scraps, craft materials, tape, glue, etc.) to provide for this challenge, giving each material a cost (for example, wood pieces cost 50¢, fabric costs 25¢, etc.) and write these on the board or on paper to hand out to the class.
Present the Introduction/Motivation content and slides to introduce students to the engineering design process (as described above). Then have students go through the steps of the engineering design process to create a locker organizer for Marisol. Inform them Marisol has only $3 to spend on an organizer, so they must work within this budget constraint. As a size constraint, tell students the locker is 32 inches tall, 12 inches wide and 9.5 inches deep. (Alternatively, have students measure their own lockers and determine the size themselves.)
As students work, ask them some reflection questions such as, “Which step of the engineering design process are you working on?” and “Why have you chosen that solution?”
Let groups present their organizers to the class and explain the logic behind their designs.
Next, distribute the case study packet and discussion sheets to the student groups. As the teams read through the packet, encourage them to discuss the differences between their design solutions and Marisol’s. Mention that in engineering design there is no one right answer; rather, many possible solutions may exist. Multiple designs may be successful in imagining and fabricating a solution that meets the project criteria and constraints.

Engineering Design Process . 2014. TeachEngineering, Web. Accessed June 20, 2017. https://www.teachengineering.org/k12engineering/designprocess

Contributors

Supporting program, acknowledgements.

This material is based upon work supported by the National Science Foundation CAREER award grant no. DRL 1552567 (Amy Wilson-Lopez) titled, Examining Factors that Foster Low-Income Latino Middle School Students' Engineering Design Thinking in Literacy-Infused Technology and Engineering Classrooms. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Last modified: October 26, 2023

Learn Engineering & Technology

Methods to Solve Any Engineering Problem

In our day to day life we came across various engineering problems. Once we face these engineering problems few questions will come in our mind like How to resolve it? What are different methods? Which is the simplest or best method?

In this paragraphs, we will discuss various methods to solve any engineering problems & their comparison with each other. There are three basic methods to solve any engineering methods.

Analytical Methods
Numerical Methods
Experimental methods

1. Analytical Methods:

The analytical method is most widely used in curriculum study as well as used by industrial designers to solve the engineering problems. It is a classical approach which gives 100 % accurate results. This approach is also referred to as hand calculations; as in this method various mathematical equations & functions are used to find output variables & derive closed form solutions. This method is mainly applicable for simpler problems like cantilever and simply supported fixed beams, etc.

Though the analytical approach is 100 % accurate, it could also give approximate results if the solution is not closed form. An equation is said to be a closed-form solution if it solves a given problem in terms of mathematical operations & functions from a given generally accepted set. For example, an infinite sum would generally not be considered closed-form.

2. Numerical Method:

When we come across more complex problems, in which both analytical and experimental methods do not work, numerical methods are driving the solutions. CAE engineers or analysts most widely use numerical methods to solve their engineering problems. This numerical method uses computational techniques through simulation software’s & large infrastructures, etc. Numerical methods do not need physical models or prototypes, it builds mathematical models to replicate real life complex problems and while doing so, several assumptions were made to simulate the analysis. Therefore, the results from this method are approximate. So, you cannot believe the results blindly and hence, sometimes sanity checks are needed to validate the simulation either by hand calculations or by physical testing, etc.

The four common numerical methods used to solve engineering problems are:

The Finite Element Method (FEM) is a popular numerical technique used to determine the approximated solution for a partial differential equation (PDE).
Applications : Linear, nonlinear, buckling, thermal, dynamic, and fatigue analysis
Powerful and efficient technique to solve acoustics or NVH problems. Just like FEA, it also requires nodes and elements, but it only considers the outer boundary of the domain. So when the problem is of a volume, only the outer surfaces are considered. Similarly if the domain is of an area, then only the outer periphery is considered. By doing so it reduces the dimensionality of the problems by one degree resulting in faster problem solving. BEM is often more efficient than other methods in terms of computational resources for problems where there is a small surface or volume ratio.
Applications : Acoustics, NVH
The FVM method representing and evaluating partial differential equations as an algebraic equations method is used in many computational fluid dynamics packages. It is very similar to FDM, where the values are calculated at discrete volumes on a generic geometry. The advantage of this method is that it is easily formulated to allow for unstructured meshes.
Applications : CFD (Computational Fluid Dynamics) and Computational Electromagnetic
It uses Taylor’s series to convert a differential equation to an algebraic equation. In the conversion process, higher order terms are neglected.
It is used in combination with BEM or FVM to solve thermal and CFD coupled problems.

Can we solve the same problem with all Numerical methods? The answer is YES, but substantial differences exist between this method in terms of accuracy, ease of programming & computational time, etc.

3. Experimental Method:

Experimental method is also known as physical testing. It is one of the most reliable methods and widely used in industry for product prototype testing.

In this method, the product or component is tested in real time operating conditions & actual measurement were reported. So in order to use this method, you will need a physical prototype of the product or structure you want to be analyzed. Only one prototype testing is not sufficient, for final outcome of analysis 3 to 5 prototype testing is required. Due to this, the experimental method is time consuming, requires expensive physical setup which results in additional cost rather than actual products.

Physical testing is performed with the help of various measuring equipment like strain gauges, different sensors, measuring devices like accelerators, etc. to calculate various parameters of the experiment. Examples: Compressor manufacturers are doing prototype testing to mitigate the vibration levels on prototypes. Here, different accelerators are placed at various point on prototype and acceleration levels are measured for operational loads.

Below images shows the simple cantilever beam problems solution by three different methods approach.

35 problem-solving techniques and methods for solving complex problems

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How do you identify problems?

How do you identify the right solution.

Tips for more effective problem-solving

Complete problem-solving methods

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

Problem-solving warm-up activities

Closing activities for a problem-solving process.

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

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

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

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

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

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

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

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

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

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

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

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

Tips for more effective problem solving

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

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

Clearly define the problem

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

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

Don’t jump to conclusions

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

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

Try different approaches

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

Don’t take it personally

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

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

Get the right people in the room

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

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

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

Document everything

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

Bring a facilitator

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

Develop your problem-solving skills

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

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

Design a great agenda

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

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

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

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

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

Design Sprint 2.0

Open Space Technology

1. Six Thinking Hats

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

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

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

2. Lightning Decision Jam

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

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

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

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

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

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

3. Problem Definition Process

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

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

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

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

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

4. The 5 Whys

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

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

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

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

5. World Cafe

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

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

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

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

6. Discovery & Action Dialogue (DAD)

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

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

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

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

7. Design Sprint 2.0

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

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

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

8. Open space technology

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

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

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

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

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

Techniques to identify and analyze problems

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

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

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

Let’s take a look!

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

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

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

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

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

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

10. The Creativity Dice

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

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

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

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

11. Fishbone Analysis

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

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

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

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

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

12. Problem Tree

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

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

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

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

13. SWOT Analysis

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

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

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

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

14. Agreement-Certainty Matrix

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

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

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

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

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

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

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

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

16. Speed Boat

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

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

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

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

17. The Journalistic Six

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

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

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

18. LEGO Challenge

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

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

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

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

19. What, So What, Now What?

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

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

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

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

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

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

20. Journalists

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

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

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

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

Problem-solving techniques for developing solutions

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

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

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

21. Mindspin

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

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

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

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

22. Improved Solutions

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

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

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

23. Four Step Sketch

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

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

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

24. 15% Solutions

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

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

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

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

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

25. How-Now-Wow Matrix

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

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

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

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

26. Impact and Effort Matrix

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

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

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

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

27. Dotmocracy

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

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

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

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

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

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

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

28. Check-in / Check-out

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

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

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

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

29. Doodling Together

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

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

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

30. Show and Tell

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

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

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

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

31. Constellations

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

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

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

32. Draw a Tree

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

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

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

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

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

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

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

One Breath Feedback
Who What When Matrix
Response Cards

How do I conclude a problem-solving process?

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

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

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

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

33. One Breath Feedback

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

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

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

34. Who What When Matrix

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

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

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

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

35. Response cards

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

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

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

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

Save time and effort discovering the right solutions

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

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

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

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

Over to you

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

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

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

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Hybrid Strategies Based Seagull Optimization Algorithm for Solving Engineering Design Problems

Research Article
Open access
Published: 27 March 2024
Volume 17 , article number 62 , ( 2024 )

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Pingjing Hou 1 ,
Jiang Liu 1 ,
Feng Ni 1 &
Leyi Zhang 1

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The seagull optimization algorithm (SOA) is a meta-heuristic algorithm proposed in 2019. It has the advantages of structural simplicity, few parameters and easy implementation. However, it also has some defects including the three main drawbacks of slow convergence speed, simple search method and poor ability of balancing global exploration and local exploitation. Besides, most of the improved SOA algorithms in the literature have not considered the drawbacks of the SOA comprehensively enough. This paper proposes a hybrid strategies based algorithm (ISOA) to overcome the three main drawbacks of the SOA. Firstly, a hyperbolic tangent function is used to adjust the spiral radius. The spiral radius can change dynamically with the iteration of the algorithm, so that the algorithm can converge quickly. Secondly, an adaptive weight factor improves the position updating method by adjusting the proportion of the best individual to balance the global and local search abilities. Finally, to overcome the single search mode, an improved chaotic local search strategy is introduced for secondary search. A comprehensive comparison between the ISOA and other related algorithms is presented, considering twelve test functions and four engineering design problems. The comparison results indicate that the ISOA has an outstanding performance and a significant advantage in solving engineering problems, especially with an average improvement of 14.67% in solving welded beam design problem.

An enhanced hybrid seagull optimization algorithm with its application in engineering optimization

Gang Hu, Jiao Wang, … Jiaoyue Zheng

MoSSE: a novel hybrid multi-objective meta-heuristic algorithm for engineering design problems

Gaurav Dhiman & Meenakshi Garg

An enhanced seagull optimization algorithm for solving engineering optimization problems

Yanhui Che & Dengxu He

Avoid common mistakes on your manuscript.

1 Introduction

Many problems in real life could be converted into optimization problems subject to complex constraints. These optimization problems are often accompanied by multiple constraints and massive computation, so that traditional optimization approaches are unable to cope with such problems [ 1 ]. To overcome the limitations of traditional methods, such as obtaining only local optimal solution and relying too much on gradient information of objective functions, researchers have proposed a new optimization method, the metaheuristic algorithm. The meta-heuristic algorithm is featured as non-derivative, high efficiency and low computation cost. It provides an efficient solution for NP-hard problems, by obtaining the optimal solution or suboptimal solution within an acceptable running time.

Due to no free lunch (NFL) theorem [ 2 ], no single optimization algorithm can be used to solve all constrained optimization problems, so it is essential to constantly explore new algorithms and improve existing ones. In recent decades, a variety of flexible and versatile metaheuristic algorithms has been put forward to solve the increasingly complicated optimization problems in various areas. For example, some standard algorithms proposed by early researchers include: genetic algorithm (GA) derived from the law of natural selection in the biosphere [ 3 ], simulated annealing algorithm (SA) based on solid annealing principle [ 4 ], ant colony optimization algorithm (ACO) inspired by the ants foraging paths [ 5 ], particle swarm optimization algorithm (PSO) proposed by simulating the foraging actions of birds [ 6 ], gravitational search algorithm (GSA) inspired by the law of gravity and Newton's second law [ 7 ], and cuckoo search algorithm (CS) which is proposed by simulating the breeding habits of cuckoo [ 8 ]. In recent years, based on previous studies, many new standard optimization algorithms have been developed. For instance, grey wolf optimization algorithm (GWO) is proposed based on the hierarchy and predation behavior of grey wolf population [ 9 ], ant lion optimization algorithm (ALO) simulates the mechanism of ant lion predation on ants [ 10 ], whale optimization algorithm (WOA) is based on the mechanism of whale rounding up prey [ 11 ], sparrow search algorithm (SSA) simulates the foraging and anti-predatory actions of sparrows [ 12 ], and butterfly optimization algorithm (BOA) is based on the behaviors of butterflies feeding on nectar and mating in nature [ 13 ].

Besides, researchers have put forward different strategies to enhance the optimization performance of the standard algorithms. For instance, introducing the mechanism of nelder-mead local search algorithm to the WOA, an algorithm named HWOANM is proposed to speed up the convergence of the WOA and to solve engineering optimization problems [ 14 ]. A new hybrid algorithm (HPSOGWO) is proposed by effectively combining PSO and GWO [ 15 ]. The negative correlation search algorithm is introduced to GSA to achieve the differentiation of search behaviors, and the test results show that the optimization accuracy of GSA is enhanced [ 16 ]. A memory-based grey wolf optimization algorithm (mGWO) is proposed to enhance the ability of balancing global and local search [ 17 ]. To overcome the disadvantages of sine and cosine search algorithm (SCA), Ref. [ 18 ] proposed the m-SCA algorithm, which introduces reverse learning strategy and self-adaptation strategy.

The seagull optimization algorithm (SOA) is proposed by Dhiman and Kumarti, simulating the migration and aggressive behavior of seagulls [ 19 ]. The SOA has the merits of simple parameter setting, easy and flexible adjustment and easy implementation. It has been studied and improved by many scholars, and been applied to different fields. Like other optimization algorithms, it also has the defects of low population diversity, easily plunging into local optimum, and weak convergence [ 20 ].

In the literature, the SOA is investigated mainly from two aspects: one is to study the optimization process of the SOA, and improve the algorithm by changing the population initialization, parameters, convergence factor or position updating method; the other is to apply the SOA to the parameter optimization of other algorithms or to some practical fields. For example, an improvement of the SOA was made by replacing the convergence factor with a nonlinear dynamic factor and using lévy flight mechanism to increase the randomness [ 21 ]. Parameters and attack angle $\uptheta $ in the SOA were adjusted and dynamic reverse learning strategy was utilized, with an application to the PID controller model [ 22 ]. The levy-flight mechanism was added to the search method of the SOA to improve the convergence rate. Then it was applied to the optimization model of the PEMFC system [ 23 ]. Ref. [ 24 ] combined the shrink-wrap mechanism of WOA with the spiral search mode of SOA, to avoid premature convergence and improve the convergence accuracy. Three strategies were introduced to improve the convergence capability of the SOA, with an application to the blind source separation [ 25 ]. The SOA was utilized to calculate the threshold points for threshold segmentation of otsu images to achieve better segmentation effect [ 26 ]. The heat transfer formula of the heat exchange optimization algorithm (TEO) was used to improve the attack formula of the SOA, providing solutions to the feature selection problems [ 27 ]. Natural selection mechanism was introduced to the SOA to avoid trapping in the local optimum, and was used to solve the dynamic optimization problems together with the unequal division method [ 28 ].

Although the researches enhanced the performance of the SOA in a way, most of them have not considered the drawbacks of the SOA comprehensively enough. At this point, this paper proposes hybrid strategies to enhance the convergence of the SOA, working on the three main drawbacks of slow convergence, poor ability to balance global and local search, and single search mode. Firstly, the helical coefficient of the SOA is improved, so that the helical radius of seagulls attacking can change with the number of iterations, and the local search ability is enhanced; secondly, an adaptive weight factor is added to the position updating method to balance the global and local search in the optimization process; finally, the chaotic local strategy is used to update the seagull position twice to prevent falling into a local optimum. In the experimental simulation, 12 test functions are utilized. Experimental comparison of solution results and convergence curves with other recent related optimization algorithms shows that the ISOA has stronger searching ability and faster convergence. In addition, four engineering optimization problems with constraints are solved by the ISOA. The results indicate that the proposed algorithm (ISOA) has strong competitiveness compared with other algorithms.

The rest of this paper is structured as follows: Sect. 2 describes the principle and optimization mechanism of the standard SOA, and analyzes the problems existing in the SOA; Sect. 3 puts forward the improvement strategies and describes the optimization procedure of the ISOA; In Sect. 4 , experimental comparative analysis is carried out to validate the performance of the ISOA; Sect. 5 applies the ISOA to four engineering design problems; Sect. 6 concludes the paper, and proposes further research.

2 Seagull Optimization Algorithm and Shortcomings

The main inspiration for the SOA is the migration and aggressive behavior of seagulls in nature. Migration is the seasonal movement of seagulls in search of the richest food source to provide sufficient energy. During migration, seagulls are supposed to keep from colliding with each other, and each seagull updates the position with the best one in the population. Then, seagulls assault prey in a spiral trajectory through the air. The migration is the global exploration phase of the SOA, while the attack denotes the local exploitation phase. The SOA is to constantly adjust the positions of seagulls to seek an optimal solution by imitating the two behaviors.

2.1 Migration Behavior

During the migration phase, the algorithm simulates how a flock of seagulls moves from one location to another. Every search agent needs to meet three conditions: avoid colliding with each other, move in the direction of the best search agent, and update the position with the best search agent.

1. Avoid collision: To avoid the mutual collision, the variable A is introduced into the algorithm to evaluate the updated position of search agents.

where ${\overrightarrow{C}}_{s}$ represents the location where the seagull does not collide with another individual; ${\overrightarrow{P}}_{s}(t)$ is the current position of the search agent; ${\text{t}}$ indicates the number of iterations; ${\text{A}}$ represents the movement behavior of seagulls in the feasible region.

where ${T}_{maxitera}$ is the maximum number of iterations, $\mathrm{t }=\mathrm{ 1,2},...,{T}_{maxitera}$ , the hyper parameter ${f}_{c}$ is used to control the size of the variable A, ${f}_{c}$ is set to 2, and the variable A decreases linearly from ${f}_{c}$ to 0.

2. Move in the direction of the best search agent: The seagulls will move to the best search agent when there is no collision between individuals.

where ${\overrightarrow{M}}_{s}$ represents the position of the seagull moving toward the best search agent; ${\overrightarrow{P}}_{best}(t)$ is the current position of the best search agent, which has a small fitness value; ${\text{B}}$ is a random number, which is used to balance global exploration and local exploitation; ${\text{rd}}$ is a random number between 0 and 1.

3. Update the position with the best search agent: After the convergence direction is determined, the seagull constantly approaches the best search agent.

where ${\overrightarrow{D}}_{s}$ denotes the distance between the search agent and the best one.

2.2 Attack Behavior

During attacking, seagulls constantly change the angle and speed, and they use their wings and weight to maintain their hovering altitude. When attacking prey, seagulls move through the air in a spiral motion. Their behavior is described below in terms of x, y and z coordinates:

where, ${\text{r}}$ is the spiral radius of each turn; $\uptheta $ is a random number in $[\mathrm{0,2\pi }]$ . ${\text{u}}$ and ${\text{v}}$ are constants; e is the base of the natural logarithm. In the standard SOA, u and v are both 1. Equations ( 6 )–( 9 ) are used to figure out the updated position of a seagull as shown below:

where ${\overrightarrow{P}}_{s}\left(t+1\right)$ is the updated position of the search agents.

2.3 Shortcomings

The SOA has the following shortcomings: (a) slow convergence. According to the position updating method of the SOA, the spiral radius ${\text{r}}$ determines the size of the search range of the seagulls. However, r is determined by the coefficients ${\text{u}}$ and ${\text{v}}$ that are constants. As a result, the search radius is too large in the later stage, causing an oscillation near the optimal solution and a failure to achieve fast convergence. (b) Poor ability of balancing global and local search. From Eq. ( 10 ), the position of the current best individual has strong influence on the position of the seagulls. The influence is supposed to vary with different stages of the algorithm. However, in the standard SOA, the weight value given to the best individual is always 1 in both early and late stages, which leads to a poor ability of balancing the global and local search. (c) Single search mode, causing a local optimal. The standard SOA has only one position updating method, as a result, search agents can only search in one way, reducing the diversity of species. This makes the algorithm susceptible to the local optimization, particularly for the multi-peak test functions.

3 Improved Seagull Optimization Algorithm

To overcome the drawbacks existing in the SOA, this paper proposes three optimization strategies: The spiral coefficient v is improved by using the hyperbolic tangent function (Tanh) to speed up the convergence; the adaptive weight factor is introduced to strengthen the ability of balancing global and local search; the chaotic local strategy is introduced to increase the diversity of search methods to improve the convergence accuracy. Finally, the solving procedure of the ISOA is provided.

3.1 Improvement of Helical Coefficient $\mathbf{v}$

From the iterative process of the SOA, when the search agent launches the attack behavior, the spiral radius r affects the size of attack range, thus has a deep impact on the optimization accuracy of the SOA. According to Eq. ( 9 ), the spiral radius r is determined by the values of the spiral coefficients u and v. In the standard SOA, the values of u and v are set to 1. As a result, the size of the spiral shape is constant and cannot be adjusted continuously with the iterations. Especially, in the later stage of the algorithm, the spiral radius cannot be decreased, even causing a failure to converge to the optimum value. At this point, the Tanh is introduced to improve the helix coefficient v.

Figure 1 is the curve of the Tanh expressed by Eq. ( 11 ), indicating that Tanh is a continuous and increasing function. Equation ( 12 ) is the improved spiral coefficient v, performed by telescopic translation of the Tanh function. From Fig. 2 , it can be seen that the value of v gradually approaches 0 with the iteration. In this way, in the early stage of the algorithm, the search agent can search with a large radius, enhancing the global exploration ability; in the later stage, the spiral radius decreases gradually, which allows the algorithm converge to the optimal solution rapidly and enhances the local search ability.

The Tanh function

Iterative curve of the helical coefficient v

where, ${T}_{maxitera}$ is the maximum number of iterations, ${\text{t}}=\mathrm{1,2},...,{T}_{maxitera}$ .

3.2 Adaptive Weight Factor Strategy

The adaptive weight factor is one of the essential factors used in optimization algorithms to balance global exploration and local exploitation [ 29 , 30 , 31 ]. As illustrated in Sect. 2.3 , the current best individual is supposed to have greater influence on the early stage of the SOA search than on the late one. So, a larger weight should be given to the current best search agent in the early stage to speed up the convergence to the neighbor of global optimal solution; in the late stage, a smaller weight value should be chosen to refrain from falling into the local optimum caused by an excessively fast convergence, thus to enhance the capability of local search. Therefore, the adaptive weight factor, expressed by Eq. ( 13 ), is utilized to improve the position updating method, as shown in Eq. ( 14 ).

where, ${\overrightarrow{D}}_{s}$ , $x$ , $y$ , $z$ and ${\overrightarrow{P}}_{best}(t)$ have the same meaning as in Eq. ( 10 ).

The iterative curve of the adaptive weight ω(t) is shown in Fig. 3 a, which indicates that the value range of $\upomega $ reduces from 2 to 0. The weight value given to the best individual in the early stage is greater than 1; the weight value in the late stage decreases rapidly and gradually approaches 0. To further prove the validity of the proposed adaptive weight factor $\upomega $ , comparison of $\upomega $ , ${\upomega }_{1}$ , ${\upomega }_{2}$ will be made in what follows, where ${\upomega }_{1}$ , ${\upomega }_{2}$ are the weight factors proposed in [ 30 ] and [ 31 ], as expressed by Eqs. ( 15 ) and ( 16 ) respectively and shown in Fig. 3 a.

Comparison of different weights $\upomega $ , ${\upomega }_{1}$ , ${\upomega }_{2}$

where, the values of ${\upomega }_{{\text{max}}}$ and ${\upomega }_{{\text{min}}}$ are the same as in Ref. [ 30 ].

When Eq. ( 10 ) in the SOA is changed to Eq. ( 14 ), we obtain a new algorithm and name it SOA- $\upomega $ . Similarly, SOA- ${\omega }_{1}$ and SOA- ${\omega }_{2}$ are the new algorithms obtained by replacing $\upomega $ in Eq. ( 14 ) with ${\upomega }_{1}$ , ${\upomega }_{2}$ respectively. Three test functions F1, F4 and F7 are randomly selected for convergence comparison of SOA- $\upomega $ , SOA- ${\omega }_{1}$ and SOA- ${\omega }_{2}$ . As can be seen from Fig. 3 b and Fig. 3 c, the SOA- $\upomega $ is nearer to the optimal solution than the other two, i.e., it has higher convergence accuracy. According to Fig. 3 d, although the SOA- $\upomega $ converges to the same optimal solution as SOA- ${\omega }_{1}$ and SOA- ${\omega }_{2}$ , it has the fastest convergence rate. In summary, the adaptive weight factor proposed is effective for balancing the global search and local search of the SOA.

3.3 Chaotic Local Search Strategy

The chaotic local search strategy uses chaotic systems to generate chaotic variables. Due to the random and uniform distribution of chaotic variables, an algorithm can perform two searches near the optimal individual, which reduces the possibility of plunging into the local optimum. Moreover, the chaotic local strategy has been introduced into other algorithms, and the experiments have proved that it can effectively improve the performance [ 32 , 33 ]. The standard SOA has only one spiral position search mode, which both reduces the diversity of the search agent and limits the search scope. It easily falls into local optimal, especially on the multi-peak functions. Therefore, to overcome the limitation of single search mode, an improved chaotic local strategy is introduced into the search process. Equations ( 17 )–( 18 ) describe the mathematical formulation of the improved chaotic local search.

where, ${X}^{\prime}(t)$ is the position of an individual, ${\text{z}}({\text{t}})$ represents the chaotic variable mapped through the chaotic system; ${\overrightarrow{X}}_{best}(t)$ is the current position of the best individual; ${\text{ub}}$ and ${\text{lb}}$ describe the boundaries of the search space; $r(t)$ is the radius of chaotic search; Eq. ( 18 ) describes how the chaotic search radius is updated, and the initial value is set as 0.01. In this paper, the strategy is adjusted and applied to the position updating method of the SOA to form a mechanism of secondary updating, as shown in Eqs. ( 19 )–( 21 ), so as to increase the diversity of the population.

where, ${P}_{s}^{\prime}(t)$ is the position of a seagull obtained by the improved chaotic local strategy, and ${\overrightarrow{P}}_{s}(t)$ is the position obtained by Eq. ( 14 ). The logistic chaotic mapping model is adopted in this paper, as shown in Eq. ( 20 ), ${\text{z}}({\text{t}})$ is a chaotic variable whose initial value is 0.152, and $\upmu $ is 4. Finally, the position ${P}_{s}^{\prime}(t)$ is compared with ${\overrightarrow{P}}_{s}(t)$ in terms of fitness. If the fitness of ${P}_{s}^{\prime}(t)$ is less than that of ${\overrightarrow{P}}_{s}(t)$ , maintain the current position ${P}_{s}^{\prime}(t)$ ; otherwise, ${P}_{s}^{\prime}(t)$ will be abandoned, as expressed by Eq. ( 21 ).

3.4 Pseudo-Code and Flowchart of ISOA

In this section, the pseudocode of the ISOA is provided based on the improvement strategies of the previous three sections. And the flowchart of the ISOA is shown in Fig. 4 .

The flowchart of the ISOA

4 Experimental Simulation and Result Analysis

Section 4 consists of five parts. Section 4.1 introduces the basic information of the test functions and the experimental environment; A comparison of the ISOA with the standard SOA and other standard optimization algorithms is in Sect. 4.2 ; In Sect. 4.3 , the ISOA is compared with other improved seagull optimization algorithms, including ISOA-1 (introducing strategy (1), ISOA-2 (introducing strategy 1 and strategy (2), BSOA [ 23 ] and WSOA [ 24 ]. Section 4.4 makes a comparison of convergence curves of all the algorithms. In Sect. 4.5 , a comparison of MAE of the algorithms is presented to further verify the optimization ability and stability of the ISOA.

4.1 Experimental Environment and Benchmark Functions

Twelve benchmark functions are utilized in this paper. F1–F4 are single-peak test functions mainly utilized to test the search ability and convergence velocity of an algorithm. F5–F12 are multi-peak test functions, among which F9–-F12 are fixed-dimensional. The multi-peak test functions have many local minimums in the search space and are used to verify the ability of an algorithm to jump out of the local minima. Table 1 illustrates the basic information about the test functions. The population size (N) for all algorithms is 30 and the maximum number of iterations ( ${T}_{maxitera}$ ) is 500. Each algorithm is independently run 30 times to obtain four indexes: the minimum (Best), the maximum (Worst), average (Ave) and standard deviation (Std). The experiments were performed in MATLAB R2020b and on a computer having an Intel(R) Core (TM) i5-7200U CPU, 8 GB of RAM, and 64-bit Windows 10 operating system.

4.2 Comparison with Standard Optimization Algorithms

We compare the ISOA with the ant lion optimization algorithm (ALO), butterfly optimization algorithm (BOA), grey wolf optimization algorithm (GWO), whale optimization algorithm (WOA) and standard SOA. The parameter settings are the same as in the corresponding references, listed in Table 2 . Table 3 shows the comparison results on 12 test functions. The value in bold indicates the best result in a row.

According to Table 3 , only the ISOA can converge to the theoretical optimum values in the F1–F4 tests among all the algorithms, moreover, it has the smallest standard deviations. This demonstrates that the ISOA has a great global optimization capability and stability in solving single-peak functions. Among the multi-peak functions, the ISOA performs the best in the F5, F7, F8, F10 and F12 tests, in terms of the four indexes. Both the ISOA and WOA converge to the theoretical optimum in the F6 test. The running result of the ISOA is slightly inferior to WOA and GWO in the F9 test. In the F11 test, although the ISOA has a slightly worse standard value than GWO, it performs better in terms of the other indexes, especially Ave. In general, the ISOA has better optimization ability and stronger stability than the other algorithms.

4.3 Comparison with Improved Seagull Optimization Algorithms

To further validate the effectiveness of the algorithm, the ISOA is compared experimentally with other improved seagull optimization algorithms, including ISOA-1, ISOA-2, BSOA and WSOA. The ISOA-1 is the algorithm improved only by the strategy proposed in Sect. 3.1 . The ISOA-2 is the algorithm improved by the strategies in Sects. 3.1 and 3.2 . BSOA and WSOA are recent related algorithms proposed in Refs. [ 23 ] and [ 24 ] respectively. Table 4 illustrates the test results of the algorithms. The value in bold indicates the best result in a row.

According to Table 4 , the ISOA has better performance than the SOA, ISOA-1, and ISOA-2 on all the functions except for F8, and ranks second on the function F8, while the SOA has the worst performance among the four algorithms. This also implies that each of the strategies proposed in this paper is valid in improving the optimization performance of the SOA. Especially, the ISOA-1 can reach the theoretical optimum in the F6 test. The ISOA-2 can converge to the optimal values in the F1–F4 and F6 tests. The ISOA can reach the theoretical optimum values in the F1–F6, F10, and F12 tests.

The ISOA performs better in the F1–F4, F7–12 tests than the BSOA and WSOA. And the three algorithms perform nearly the same in the F6 test. In the F5 test, the ISOA converges to the theoretical optimal solution in terms of the index Best, while ranks second in the Ave and Std. On the whole, the ISOA has strong competitiveness in search ability and stability.

4.4 Comparison of Convergence Curves

The convergence curves of all algorithms in the F1–F12 tests are shown in Fig. 5 . From Fig. 5 , the ISOA has the fastest convergence rate and the highest convergence accuracy in the F1–F4 tests. In particular, the ISOA can converge to the optimum within about 380 iterations in the F1 test. In the F5–F12 tests, the ISOA still converges dramatically faster than the comparison algorithms except for function F8. In the F8 test, the ISOA performs slightly worse in the early part of the test, but better in the later part. Especially in the F10–F12 tests, ISOA converges to the optimal value within less than 200 iterations. It can also be found in Fig. 5 that the comparison algorithms such as BSOA and WSOA easily fall into the local optimums, and their capability to jump out of the local optimums is worse than ISOA’s. All above shows that the ISOA has faster convergence speed and stronger global exploration and local exploitation ability.

Convergence curves of all algorithms in the F1–F12 tests

4.5 Sorting by Mean Absolute Error (MAE)

Now the ISOA is compared with the published algorithms from Sects. 4.2 and 4.3 in terms of MAE. The MAE is an indicator used to describe the gap between the actual optimum value of an algorithm and the theoretical optimum value, which is expressed as follows.

where, ${\text{N}}$ is the number of test functions selected; ${{\text{F}}}_{{\text{i}},{\text{best}}}$ is the average value of the results acquired by running algorithms for 30 times on the i-th test function; and ${\overline{{\text{F}}} }_{{\text{i}}}$ is the theoretical optimal value in the i-th test function. Table 5 shows the sorted MAE values. The ISOA has the smallest MAE value, i.e., the optimization result of the ISOA is the closest to the theoretical optimal value. This further indicates that the ISOA has better convergence accuracy.

5 Engineering Applications

This section verifies the advantages of the ISOA in optimizing four engineering design problems with different complexities. Two types of algorithms are selected for comparison tests, including the standard algorithms and improved algorithms. All algorithms are run independently 30 times, with the population size and the number of iterations set to 30 and 1000 respectively, and all the comparisons are based on the best-case results.

5.1 Three-bar Truss Design Problem

The three-bar truss design problem [ 34 ] is a typical problem in engineering applications, whose optimization objective is to minimize the volume of the truss structure under certain conditions. Figure 6 shows the model structure diagram of the three-bar truss. The model uses two variables ${A}_{1}$ and ${A}_{2}$ to modify the cross-sectional area of the rods. The cross-sectional area of the two sides is ${A}_{1}$ , and the cross-sectional area of the middle rod is ${A}_{2}$ . The objective function and constraint conditions are as follows:

Model structure diagram of the three-bar truss

Consider $x=[{x}_{1} {x}_{2}]=[{A}_{1} {A}_{2}]$ ,

where, $0<{x}_{1},{x}_{2}<1$ , other parameters: $l=100$ cm, $p=2KN/{cm}^{2}$ , $\delta =2KN/{cm}^{2}$ .

The ISOA is compared with other algorithms proposed in recent years, including ALO [ 10 ], WOA [ 11 ], AOA [ 35 ], SCA [ 18 ], HHO [ 36 ], MFO [ 37 ], m-SCA [ 18 ], MMPA [ 38 ], GSA-GA [ 39 ], AGWO [ 40 ], BWOA [ 41 ], NCCO [ 42 ], PSO-DE [ 43 ], DEDS [ 44 ] and ESOA [ 20 ]. As shown in Table 6 , the convergence result of the ISOA is the best, and the ISOA has stronger optimization accuracy. The optimal solution is 263.8956 and the corresponding optimum variable is $x=[0.78812, 0.4098$ ]. Figure 7 gives the convergence curve of the ISOA, indicating that it takes only 400 iterations to converge to the optimum value, and the ISOA has a fast convergence speed.

Convergence curve of the ISOA in solving three-bar truss design problem

5.2 Pressure Vessel Design Problem

The optimization objective of the pressure vessel design problem [ 45 ] is to minimize the total manufacturing cost consisting of material cost, molding cost and welding cost of cylindrical vessel under four constraints. Figure 8 shows the structure diagram of the pressure vessel model. This design problem involves four optimization variables, which are the thickness of the shell ( ${T}_{s}$ ), the thickness of the side of the head ( ${T}_{h}$ ), cylinder radius ( ${\text{R}}$ ) and the length of the cylindrical shell ( ${\text{L}}$ ), where R and L are both continuous variables. The specific mathematical expressions are as follows:

Structure diagram of the pressure vessel model

Consider $x=[{x}_{1} {x}_{2} {x}_{3} {x}_{4}]=[{T}_{s} {T}_{h} R L]$ ,

In tackling this problem, the ISOA is compared with the standard optimization algorithms, including PSO [ 6 ], SOA [ 18 ], GWO [ 9 ], WOA [ 11 ], AOA [ 35 ], and SOS [ 46 ], and other recent related improved optimization algorithms, including ESOA [ 20 ], WSOA [ 24 ], RCSA [ 47 ], IDARSOA [ 48 ], TLMPA [ 49 ], EEGWO [ 50 ], hHHO-SCA [ 51 ], MMPA [ 38 ] and ASOINU [ 52 ]. According to Table 7 , the lowest manufacturing cost for the pressure vessel solved by the ISOA is 5805.7158, and the corresponding optimum variable value is $x=[0.7735698, 0.3679545, 41.59672, 182.9594]$ . The ISOA has the best result among all the algorithms, showing its strong competitiveness in searching optimal solution. As shown by the convergence curve of the ISOA in Fig. 9 , the ISOA converges rapidly in the initial stage, jumps out of the local optimum within only about 400 iterations, and reaches the optimal solution within only about 480 iterations.

Convergence curve of the ISOA in solving pressure vessel design problem

5.3 Welded Beam Design Problem

The optimization objective of the welded beam design problem [ 53 ] is to minimize the manufacturing cost subject to seven constraint conditions. This design problem involves four variables, which are weld thickness (h), cleat length (l), beam height (t) and beam thickness (b), as illustrated in Fig. 10 . It also involves four functions $\uptau $ , $\upsigma $ , $\updelta $ , ${ P}_{c}$ , which denote the beam bending stress, shear stress of the welded beam, the deflection at the end of the beam, and bar buckling load respectively. The following is the mathematical model of the problem:

Structure diagram of the welded beam model

Consider $x=[{x}_{1} {x}_{2} {x}_{3} {x}_{4}]=[h l t b]$ ,

where, $0.1\le {x}_{1}\le 2, 0.1\le {x}_{2}\le 10, 0.1\le {x}_{3}\le 10, 0.1\le {x}_{4}\le 2$ ,

The ISOA is compared with the algorithms proposed recently, including HWOANM [ 14 ], WSOA [ 24 ], IDARSOA [ 48 ], BFOA [ 54 ], hHHO-SCA [ 51 ], GSA [ 7 ], SCA [ 18 ], SBO [ 55 ], HHO [ 36 ], T-cell [ 56 ], HEAA [ 57 ], Random [ 58 ] and Coello [ 59 ], and the experimental results are illustrated in Table 8 . The optimization result of the ISOA is superior to the others except for HWOANM, with an average improvement of 14.67%. Although the ISOA ranks second in optimization result, it converges to the optimal solution within only about 500 iterations, while the HWOANM takes 2300 iterations [ 14 ], as shown by the convergence curve in Fig. 11 . The ISOA also has advantage in addressing the complex engineering problem.

Convergence curve of the ISOA in solving welded beam design problem

5.4 Speed Reducer Design Problem

The speed reducer design problem [ 60 ] is a classical engineering optimization problem. The optimization objective of this problem is to minimize the weight of a reducer under inequality constraints. The constraints are with respect to bending stress of the gear, surface pressure, lateral deflection of the shaft and pressure in the shaft, and involve 7 optimization variables, including the width ( ${\text{b}}$ ), the tooth module ( ${\text{m}}$ ), the number of teeth in the pinion ( ${\text{p}}$ ), the length of the first shaft between the bearings ( ${l}_{1}$ ), the length of the second shaft ( ${l}_{2}$ ), the diameter of the first shaft ( ${d}_{1}$ ) and the second shaft ( ${d}_{2}$ ), where the number of teeth ( ${\text{p}}$ ) is an integer. The model structure diagram is in Fig. 12 . The mathematical description of this problem is shown below:

Structure diagram of the speed reducer model

Consider $x=[{x}_{1} {x}_{2} {x}_{3} {x}_{4} {x}_{5} {x}_{6} {x}_{7}]=[{b m p {l}_{1} {l}_{2} d}_{1} {d}_{2}]$ ,

where, $2.6\le {x}_{1}\le 3.6,$ $0.7\le {x}_{2}\le 0.8,$ $17\le {x}_{3}\le 28,$ $7.3\le {x}_{4}\le 8.3,$ $7.3\le {x}_{5}\le 8.3,$ $2.9\le {x}_{6}\le 3.9,$ $5.0\le {x}_{7}\le 5.5$ .

The ISOA is compared with both the standard algorithms and the latest algorithms, including PSO [ 6 ], SOA [ 18 ], GWO [ 9 ], AOA [ 35 ], SHO [ 61 ], CA [ 62 ], ESOA [ 20 ], hHHO-SCA [ 51 ], IAFOA [ 63 ], IPSO [ 64 ], IDARSOA [ 48 ], QOCSOS [ 52 ], ASOINU [ 52 ] and ISCA [ 65 ]. The solution to this design problem by the ISOA is 2973.91750, the optimum variables is $x=[3.40385, 0.7, 17, 7.74585, 7.76495, 3.32186, 5.25780]$ . From Table 9 , the ISOA provides lighter weight of the speed reducer compared to other algorithms. Moreover, the ISOA has fast convergence speed, converging to the optimal solution within only 170 iterations, as shown by the convergence curve in Fig. 13 . The ISOA has strong competitiveness in solving complex engineering design problems.

Convergence curve of the ISOA in solving reducer design problem

6 Conclusions and Future works

This paper presented an improved seagull optimization algorithm named ISOA, by combining a variety of improvement strategies to overcome the drawbacks of slow convergence, poor ability to balance global and local search, and single search mode. Firstly, the strategy of adding spiral factor to the spiral radius in the attack stage enables search agents to adjust the search radius with the increase of the iteration number, so that the ISOA can not only converge quickly in the early stage but also avoid missing the optimal solution in the late stage caused by an excessively large radius. The second strategy of utilizing dynamic adaptive weight factors adjusts the proportion of best individuals to achieve equal emphasis on global exploration and local exploitation. Finally, the chaotic local search strategy is added to update the algorithm twice, expanding the search scope, and improving the capability to jump out of the local optimal.

The comprehensive comparative experiments on 12 benchmark test functions, including single-peak and multi-peak functions, show that the ISOA has greatly enhanced the optimization accuracy and convergence rate of the original SOA, and has strong competitiveness among the recent related optimization algorithms. Moreover, the ISOA solves 4 engineering optimization problems of the three-bar truss design, pressure vessel design, welded beam design and speed reducer design. The experimental results show that the ISOA generally offers better solutions, and has a fast convergence speed.

In future works, the proposed ISOA and strategies will be applied to solve optimization problems in practical fields, such as UAV path planning, inventory control and resource allocation. For instance, one of our ongoing work is to apply ISOA to multi-scenario and multi-obstacle UAV path planning, and we have made some progress. In addition, another research direction worth being further explored is an application of ISOA to data science such as differential privacy.

Data Availability

Not applicable.

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The authors thank the anonymous reviewers for their thoughtful suggestions and comments.

This work was partly supported by the National Natural Science Foundation of China under Grant Nos. 12371508, 11701370.

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Pingjing Hou, Jiang Liu, Feng Ni & Leyi Zhang

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PH and JL provided the main concept of this work. PH and FN wrote the main script text. PH and LZ made the experiments. All authors reviewed the manuscript. Corresponding author: Correspondence to JL.

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Hou, P., Liu, J., Ni, F. et al. Hybrid Strategies Based Seagull Optimization Algorithm for Solving Engineering Design Problems. Int J Comput Intell Syst 17 , 62 (2024). https://doi.org/10.1007/s44196-024-00439-2

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