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I am an Art of Problem Solving Drop-out.

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By Jenn in Mo , September 21, 2010 in High School and Self-Education Board

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We're a mathy family here, so I'm feeling kind of like a failure here, but I closed the book today and said, "Enough." My math-loving son was diligently struggling through hours of math a day and getting nowhere. It took nearly two months to get through two chapters and I know he couldn't pass any review tests. He understands algebra, but this is....different. My husband and I are struggling to understand half of the problems ourselves...and this is our *thing*, ya know?

I haven't heard a negative word about AoPS yet, so it's kind of embarrassing to admit, but I just do not "get" some of these questions. Anyone that gets it feel like helping to clear the mud on this one?

2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?

The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).

My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:

My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

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2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?   The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).   My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:   My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

(Okay LOL on the Frankie Avalon - I love that song!)

But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Have you considered the classes? I know it's a lot of money for a maybe (although I think you have three weeks to bail and get a refund) but I find that there are things I should know but can't always blurt out right when DS needs them... and I like the idea of having someone else who is prepared to do it and other kids who can add to the discussion. Even one math jam left me going "wow - this guy rocks"... :) DS is starting two AoPS classes this year, so I'm not really speaking from experience yet, but I know I'd be stuck going "uhhhhhh...." even on things I really should be able to get. It is hard stuff, and not the "I can teach this in my sleep" of other algebra and geometry programs. I've tutored those for years, but AoPS really is quite a bit more.

8filltheheart

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.

I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.

My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)

FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

ETA: Thought I should share that if I had known about AoPS umpteen yrs ago, I would have wanted my oldest to try them. I do think they stretch mathematical comprehension beyond the norm......basing that on the fact that my 14 yos now talks some strange language that I don't have a clue about what he is saying. All that said, I **know** that the rest of my older kids would say no way. (as a matter of fact my 11th and 6th graders did!! )

Guest Cheryl in SoCal

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.   I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.   My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)   FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

:iagree:I haven't used it yet but have been looking through the Intro to Counting and Probability and a bit of the Intro to Algebra and from what I have seen I wholeheartedly agree. No math curricula is the answer for every student but IMHO AoPS is going to be a good fit for the minority of students. One of my ds's would DIE if I had him use AoPS and another I think will like it because he thinks that way, though we'll only know after we try it.

I had a look at the algebra and geometry and found the layout visually confusing, which is kind of weird since in some ways it is not as busy or distracting as textbooks in general. But it is packed solid, and hard to read as a text, if that makes any sense. And algebra is my own strength. My dd went for Discovering Algebra instead.

The geometry book struck me similarly as user-unfriendly, but that may be again just my visual reaction to such a dense text.

However, that similar packing did not bother dd with the data and probability books, which she greatly enjoys. So maybe it depends on the kid (and grown-up) as well as on the area of math involved.

(Okay LOL on the Frankie Avalon - I love that song!)   But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Ha! Lightbulb! Thank you! Regardless of what x is, the expression will equal the same thing. You're speaking my language. The book is not. :tongue_smilie:

I am relieved to hear these responses. He loves math and enjoys a challenge, but I couldn't at all say that he is passionate about math. I had forgotten that Foerster was on my list of maybes before I heard that AoPS was the crème de la crème of algebra texts. So....NEM or Foerster....thoughts?

I know absolutely nothing about NEM, so I can't offer any insight there. I have used Foerster with all of my older kids, including my ds that now uses AoPS. He used Foerster for both alg 1 and alg 2. His first AoPS courses were Intro to Counting and Prob and their alg 3 courses online and he owns several of their other "off normal track" math books that he works through on his own.

My oldest used Foerster and it more than prepared him for university level cal and an engineering major.

I'm not sure I can explain the differences very easily. Foerster is a solid math program and they absolutely learn the math skills they need to perform high level math problems, etc. No question.

AoPS helps them process math in a completely different fashion. Ds goes around "discovering" concepts and "proving" things like why x equals some bizarre thing or how an indefinite number of fractions add.......I don't know......I just nod my head!!! :lol: He thinks he wants to double major in astrophysics and math. For him it is a match. I have only ever met 2 other people that think that way.......one is now majoring in math and the other has a phD in math. For the rest of mere mortals, I think we are fine w/o it!

In The Great White North

Dc used/are using Foerster. They arde very solid math books. I have the AoPS Problem Solving, Counting & Probability and Number Theory books. Dc don't like them. They are not straight forward. Dc like to have an example, an explanation and get the problems done. Foerster does this for them. AoPS does not.

AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I've seen but not used Foerster, so I can't compare... but NEM is good challenging work. Most of it is more straightforward than AoPS (examples first, exercises second) but there are Challenge sections and Mathematical Investigations that could be frustrating in the same ways AoPS can be. They do (in those sections) throw you in and expect you to dig yourself out. And to some extent they leave some gaps in the regular explanations too -- nothing huge, but enough that sometimes the student (or the teacher) has to connect the dots.

C_l_e_0..Q_c

I'm in the opposite situation. We're a NEM drop-out. I'm so much happier as a mom with AoPS *and* the online class. Someone else is teaching math to my math kid!

(and I"m good at math too, I did math all the way to a Master's level, although it was not my major - I'm an engineer, we need math )

My son is truly having a blast with the online class. He did Algebra 1 over the summer. The kids at the summer pool who learned he was doing a math class assumed it was remedial, until they saw a problem! LOL. Sweet nerd revenge. :)

The math classes are totally worth the cost. Plus they force the kids to move forward, and not spend 2 months on 2 chapters. In 15 weeks, they did 14 chapters (possibly 13, I'm not sure) so the interest is kept high.

at the beach

I agree that AoPS is for people who enjoy playing around with math. I think I am one of those who said something along the lines that more people can be successful with it than the top few percent. But I think I also said that sometimes the solutions/explanations gloss over things or skip a step, assuming readers will see that step in the problem when they might not. I felt it could be used by more than just the top few percent because I think that kids often underestimate their abilities. I also agree with the pp that even kids in the top few percent who can do it may not want to do it because it does take a lot of time and puzzling.

I was terrified of math from 8th grade on. I never took math in college but took logic to avoid it. When I signed my daughter up for AoPS this summer, I decided to give the book a try. I did the first three chapters. I wanted to help my daughter if she needed it, but she was the one helping me :001_huh: and the class moved so fast that I couldn't keep up.

The problem that you mention, to me, looks like it is asking you to make the 2x the same as the 3x so the only thing that works is 3/2. How do you know it will cancel? I think you know that because it's a subtraction problem. Maybe I am oversimplifying this.

Maura in NY

NEM is an integrated program - Algebra/Geometry/Trig and rolled together. It's challenging, in a good way, but it doesn't fit the typical Algebra/Geometry/Algebra II/Trig-PreCalc/Calc track.

FWIW, my ds used NEM 1 in 7th, thinking he would stick with it for 4 years. Instead, we switched. He didn't use Foerster's Algebra I, but he did use Foerster for Algebra II and PreCalc. Great real world application problems.

We used AoPS for several months and also found some of the problems strangely worded. But, our biggest problem with AoPS was not enough repetition.

We are using Foerester's Algebra now. It has more than enough problems for us, but it's much easier to eliminate problems than try to add more in.

Miss Marple

Not a drop-out yet, but we are doing AoPS very slowly and I've added NEM2 back as the primary math. Ds is still working through the geometry book (45 minutes per day), but we've put the algebra book away because he already did LOF Beginning Algebra and the NEM should cover the bases thoroughly.

I'm finding that hormones cause my son to be less inquisitive and less dedicated right now. He does great with math, so I chose the other program I love - Singapore NEM, and it is going very well.

If we never finish the AoPS books that will be OK because NEM is the primary curriculum (which we will finish). I know that any exposure ds has had with AoPS has helped fine tune his mathematical abilities.

Well, I was in a serious minority (a group of 1, me ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it

:iagree: I cannot imagine how frustrated a student and a parent would be trying to work through this program and thinking that it is accessible for all students!

The problems in NEM can be plenty challenging at times. My son did NEM 1 before switching to AoPS.

Regarding AoPS Algebra 1, they introduce Algebra 2 concepts, too. That doesn't make it easier! Ds thought the middle of the text was difficult and often trudged through the problems, but then he flew through the last few chapters. He thought they were easier, even many of the challengers. I don't know if they really were, or if he just started to get the hang of the text and problems.

When it comes to math, it's so important to pick what works and then bite off and chew well at a pace that's suitable for the student. I wouldn't hesitate to step back and reconsider if something wasn't working.

  AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I agree entirely with the bolded.

If anybody is deciding whether to go with the AoPS classes, there's a Math Jam on the subject this afternoon/evening (7:30 Eastern, I believe). It will include sample questions from the class, so you can try them out.

Math Jams are free informational sessions, usually hosted by Richard Rusczyk, who is always responsive to questions. If he can't answer your question within the confines of the session, he'll likely invite you to e-mail him and he'll help you decide whether a class is right for your family.

We're mathy. . . and we're AoPS drop outs as well.

Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

The exploration and problem solving was interesting. I rather enjoyed it. But, I just *don't have time* to do it right now.

We dropped it after a few chapters -- which were successful but slow b/c I just couldn't make myself find the time to do it often enough to keep up w/ dd's needed pace.

So sad. . .

Now y'all are scaring me. My dd loved the AoPS Number Theory book so much this summer, I bought Alg I for her (but we don't start for another year...) I had been planning on Foerster's, so I've got both ready to go (well, I still need that pesky Foerster's Solutions manual...)

We're mathy. . . and we're AoPS drop outs as well.   Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

Dd and I worked through all the examples together, but then she did the problem sets pretty much on her own (except I let her skip the challenge problems at the end of the chapters). Was your dd just needing you to sit with her for the "lesson" part, or also the problem sets? I didn't mind working through the lesson with her... I actually learned a lot myself and had fun... :tongue_smilie: For some reason I really like the layout and presentation of these books... OTOH the Lial's layout really drove me crazy when I first saw it (I'm getting used to it now that my other dd's working through BCM).

But now you guys are intimidating me a bit... :001_huh: I do think the Algebra text looks more challenging than the Number Theory, but I'm hoping after she's gotten through Singapore DM1, she'll be preparerd (or I guess we'll do Foerster's after all...)

I found the solutions manual for Foerster this afternoon! I finally called the publisher and it turns out that they renamed the book and it is now called Classic Edition instead of Foerster's. The ISBN # is 0201861003. I can't find it used anywhere, but it is available new at pearsonschool.com.

I'm still torn though. I loved Singapore so well and would like to try going on with NEM. But I don't want to commit to it being his math for all of high school. I hear that Foerster does an excellent job with proofs, which I think are important. I may end of tossing a coin before this is over.

This is exactly what I was trying to express in the other thread! It's just as much about the approach as the difficulty, imo. Yes, you need to be a good math student, but I dont' think you need to be a TOP math student. Not in the sense of the top 2% to 3%, at any rate. I can't imagine my dd would score anywhere near that high, but she loves AoPS.

And you can be a top math student, but if you hate this approach, it will be very hard to persevere and succeed.

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Art of Problem Solving Math Books Review

What is art of problem solving.

Founded in 1993 by former USA Math Olympiad winner Richard Rusczyk, Art of Problem Solving (AoPS) is a company that produces rigorous math instruction courses and products that can help outstanding math students develop a more thorough understanding of math concepts, as well as help prepare them for success in math competitions. 

From textbooks to online classes to physical learning centers, AoPS offers a variety of educational products and services that can help challenge kids, deepening their knowledge and strengthening their mathematical thinking.

AoPS Math textbooks

Art of Problem Solving has created a series of textbooks for middle and high school math textbooks that are designed to give outstanding math students a deeper and more rigorous curriculum in math. 

Originally designed to help talented math students prepare for competitions, over the years AoPS’s textbook line has expanded to offer full curriculums in middle and high school math courses, and their problem-based and rigorous approach to math has made them very popular with parents across the world as a top enrichment option. 

What Grades and Math Subjects does AoPS Math cover?

Art of Problem Solving textbooks cover middle and high school math, as well as competition prep.  

Generally speaking, the AoPS math textbooks can be broken down into two curricula- introductory and advanced – that roughly correspond to most middle and high school math programs (in terms of overall scope, that is). 

Parents of younger math enthusiasts should note that Art of Problem Solving covers elementary school math (Grades 1-6) in their Beast Academy series, which you can read about in our review .

Introductory Curriculum (Middle School)

Advanced Curriculum (High School) 

When taken as a whole, Art of Problem Solving’s math textbooks cover the topics included in most US Math curricula, as well as touching on a few topics that aren’t usually covered in most public high school programs. 

That said, the point isn’t really to get kids learning college level math or a curriculum beyond high school math, but instead to get students to develop their problem solving skills and develop more creative and flexible mathematical thinking, to get them to recognize and appreciate different approaches to problem solving, as well as getting a better understanding of the why of math, rather than just focusing on how to compute problems. 

As such, AoPS’s curricula tend to go deeper into your typical middle and high school math topics, letting kids examine concepts more rigorously, more thoroughly and with more challenging problems than they would otherwise be able to do in other math courses. 

Art of Problem Solving Contest Prep

In addition to their more academically-focused textbooks, Art of Problem Solving also offers a variety of books designed to further enrich exceptional students or help with preparing for math contests and Olympiads.

These books generally tend to work on developing stronger problem solving skills, going far deeper into various concepts and exploring far more challenging questions and problems, while introducing various approaches for understanding and solving them quickly and effectively.

Geared more for gifted enrichment and contests preparation, each of these books tend to go over a greater variety of concepts and topics, touching on concepts in Geometry, Algebra, Number theory and more, and aren’t really bound to any linear curriculum. 

In addition, the problem sets, geared as they are to helping students prepare for national tournaments and contests, are far more challenging and in-depth than would be expected of even an advanced middle or high school course.

For these reasons we don’t usually think this series is where parents should necessarily start off when working on math at home, but in our experience we do feel they are great supplements to the main textbooks and can be excellent for enrichment purposes and preparing for contests.

How Art of Problem Solving Teaches Math

Aops pedagogical approach.

Art of Problem solving is a big believer in teaching through solving problems. 

The books consequently include a wide variety of problems, many of which kids will have never encountered before.

In fact, some come directly from various math competitions such as:

• The American Mathematics Competitions (AMC)  
• The Harvard-MIT Math Tournament

The general idea is that by getting kids to work through problems themselves, and more importantly discovering how to solve certain problems, kids will develop a deeper understanding of the material. 

As a result, AoPS Math textbooks are quite problem set heavy.

Explanations of each concept are quite short and to the point and are followed by a good deal of exercises for students to try out on their own.

When introducing these textbooks, parents should expect that kids will have to think things through a bit more and work out the answers themselves without a lot of hand holding or spoon feeding, and that there will be a heavier emphasis on logic and proof than other curricula. 

All this really drives home Art of Problem Solving’s place as a resource for outstanding or talented math students who don’t need a lot of time or explanation to grasp the material. 

Consequently, students who are less adept at math may find the instructions a little too short and too quick and may need extra help in order prevent getting frustrated by skill and knowledge gaps as the exercises come rolling in

Lesson structure

Regardless of the book in question, Age of Learning’s lessons tend to follow a particular format.

The books are made up of several chapters, each of which covers a particular topic within the subject and contains several sections. 

Each section is then typically broken down into various related concepts, an overview of the types of problems kids may come across (both common and uncommon) and often the various factors that can affect outcomes.

In Introduction to Algebra, for example, when discussing multivariable linear equations, the chapter is divided up into an introduction, a discussion of substitution, elimination, some word problems, common and uncommon problem sets, different variables and so on.

As kids go through their lessons, they are given lots of examples to try and lessons tend to work through some of them step-by step in a fairly in-depth and rigorous manner to demonstrate concepts. 

Sections typically end with a variety of exercises for that section and, at the end of each chapter, there are review and challenge problems. 

Review problems go over and test what the student has learned with similar problems, while challenge problems go a step further and test mastery of the material with far more challenging questions. 

If kids get stuck, there are always hints and solutions that are helpfully included in the back of the book ( no cheating !)

Look and feel 

As you might expect from a problem solving and word problem-heavy methodology, these textbooks contain lots of typical math diagrams and pictures floating about to go along with and illustrate the word problems.

AoPS textbooks also tend to have a lot of floating boxes that highlight important information for kids, including: 

• Pointing out various strategies they can take on given concepts or problems
• Offering extra work
• Giving extra information
• Even offering “bogus” solutions that point out the most common mistakes made by students when solving a problem

Despite its rigor, Art of Problem Solving does its best to keep its material from becoming too dry and boring, which we appreciate. 

The books are written in a very casual tone, which makes it feel as if a math-whiz friend were explaining the material rather than a textbook. 

There are also a good deal of amusing and interesting examples and concept demonstrations sprinkled throughout, sometimes even involving sly pop culture references (some of which may go over kids heads, but parents will appreciate).

Does this approach really work?

Due to its philosophy and the way it teaches, we feel the Art of Problem solving takes more or less a constructivist/Problem Based Learning approach to teaching math where, instead of receiving formal lectures about math, students build up their own knowledge and skill by working through and solving various problems.

This learner-centric approach to teaching math and science actually has been linked to positive outcomes when teaching math and science , fostering greater problem-solving skills, improving self-motivation and encouraging creative and critical thinking skills as they relate to mathematics. 

Past customers have also reported that the series challenges their students pretty thoroughly, increasing the depth of their knowledge on relevant subjects and increasing their speed at solving difficult-math problems, sometimes dramatically. 

It is perhaps unsurprising, then, that the Art of Problem Solving curriculum is often used in honors math classes across the US. 

Some Drawbacks to Art of Problem Solving Textbooks and Curriculum

Can be time consuming.

Due to its focus on doing exercises, exploring concepts and working through problems to gain a better understanding of the subject matter, Art of Problem Solving can take a little more time to work with than some other programs.

This can be particularly true as AoPS tends to use far more challenging questions than kids are used to, some of which are in formats they haven’t seen before. 

While great for learning, this approach isn’t exactly a time saver. It’s not uncommon, for example, for parents to report spending up to 45 min (or more) each day on math (in addition to other homework). 

Can be tricky to jump into from another curriculum

With its particular approach and pedagogy, as well as its more rigorous approach to mathematics and problem solving (including the use of proofs), Art of Problem Solving can be somewhat tricky to get used to if you jump into it from another curriculum. 

Because math is a cumulative process, kids who begin Art of Problem solving without having at least reviewed some of the foundational material in previous books can find themselves lost or slowed down by skill and knowledge gaps they didn’t realize they had. 

Helpfully, the AoPS website does have free, printable diagnostic assessments for each book to help parents determine if their kids are at the right skill level. 

Discovery approach can frustrate some learners

Despite the fact that Age of Problem Solving’s approach has been shown to get results and improve the mathematical thinking and skills of talented math students, sometimes it just isn’t the right approach for the student. 

AoPS often requires students to play around with numbers and concepts and discover missing information themselves. 

Some students, even really talented students, can get frustrated by this approach and may prefer a more straightforward, traditional math course where they can get down to computation and see their results more quickly. 

Who is Art of Problem Solving For?

Overall, we think Art of Problem Solving is a great resource for parents and kids looking for a far more thorough, challenging and enriched math program.

It is an ideal course for students who demonstrate an aptitude for math and are looking to deepen and strengthen their math skills with more challenging grade-level material.

We think AoPS textbooks can be particularly good for students interested for more rigorous preparation for math-heavy STEM subjects in university , where their greater focus on problem solving, proofs and logic skills will be a strong asset, such as with physics, engineering and even computer science,

We also think that Art of Problem solving’s textbooks and methodology can be an excellent base material for students interested in or preparing for math contests and olympiads (AMC 10, AMC 12, MATHCOUNTS and the like), particularly their Contest Math Prep Series, as they promote creative approaches to problem solving and strengthen mathematical thinking that kids can use when faced with new problems.

Who is Art of Problem Solving Not Great For?

That said, Art of Problem Solving textbooks are obviously not for every student. 

These books are not the best curriculum for kids who are struggling with math concepts as AoPS math is primarily aimed at enriching math study. 

AoPS math goes far deeper into the material with far more rigor, exploring various high school and middle school math topics at a more advanced level and with more challenging problem sets, while emphasizing multiple approaches to problem solving and flexibility when approaching new math problems. 

Struggling students, while they often can benefit from learning the why’s behind math, can usually spend their time better by reviewing the fundamentals and practicing basic strategies, as well as by working on more targeted skill development with programs like IXL and Khan Academy .

Similarly, we don’t feel that AoPS textbooks are really the best resource for preparing for the SAT and other timed standardized tests where answering speed and efficiency (and test taking strategies) can be far more effective when it comes to success than gaining a deep understanding of concepts and working through problems.

In these instances, kids are better served through specific standardized prep programs that will work with them on developing their proficiency at solving very particular types of questions. 

Finally, AoPS textbooks are also not the best solution for kids looking to explore college level math as, despite its more challenging nature, AoPS math goes deeper into middle school and high school math topics (algebra, geometry, number theory, single variable calculus), rather than beyond it.

Price: How much do AoPS Textbooks Cost? 

The price of AoPS math textbooks really depends on the particular book and subject you’re interested in. 

Generally speaking, though, each book costs between $45 and $70, which is roughly the same as the average middle or high school textbook .

The length of each book varies, however, from just under 300 pages of instructional material in some cases to well over 700 in others.

Unlike many other middle and high school textbooks, however, these are designed to serve as a complete curriculum for each topic as every book contains instructional material as well as hundreds of practice problems, hints, and a step-by-step solution guide that itself is usually a couple hundred pages long as well. 

Bottom Line:

If you have a talented middle or high school math student and you’re looking for ways to nurture their excellence, Art of Problem Solving’s math textbooks might be right for you.

Although certainly not for everyone, with their challenging curriculum and in-depth exploration of math concepts, AoPS can foster better problem solving skills, stronger analytical ability and improved creative and critical math thinking, all of which can help students take their math skills to the next level.

About the Author

David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.  

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The Art of Problem Solving Math

The Art of Problem Solving (AoPS) math courses for grades five through twelve were designed for high-performing math students. The publisher says on their website ,

 We present a much broader and deeper exploration of challenging mathematics than a typical math curriculum and show students how to apply their knowledge and problem-solving skills to difficult problems. We help students learn the critical problem-solving skills necessary for success at mathematics competitions (such as MATHCOUNTS and the AMC), top universities, and competitive careers.

Their courses cover much more than typical math courses for middle through high school. They have courses that cover the standard sequence at advanced levels, plus other courses that take students deeper into the math needed for physics, engineering, computer science, and other math-based careers.

 AoPS students often work a few years ahead of other students, which means that capable fifth or sixth graders might start with Prealgebra . Note that AoPS is the publisher of Beast Academy math courses for grades one through five, and those courses prepare students to move right into AoPS Prealgebra.

Courses and Format Options

AoPS lists five courses as part of their “Introductory Curriculum” for students up through tenth grade : Prealgebra, Introduction to Algebra, Introduction to Counting & Probability, Introduction to Geometry , and Introduction to Number Theory . Their “Intermediate Curriculum” for advanced high school students includes Intermediate Algebra, Intermediate Counting & Probability, Precalculus , and Calculus . Even so, students in a traditional program can still use these courses following a more typical timeline and concentrating on the required courses.

The website page for each course has two free diagnostic tests (PDFs) that help determine whether a student has the prerequisites for the course or whether they already have mastered what the course covers. These tests are accessed by clicking on “Are You Ready?” and “Do You Need This?” on each course’s description page.

Students do not need to complete all books in the series, but if they start the series in sixth grade, they should be able to complete most of them. Students who want to participate in math competitions might also be interested in AoPS books written specifically for that purpose: Competition Math for Middle School; the Art of Problem Solving, Volume 1: the Basics ; and the Art of Problem Solving, Volume 2: and Beyond . (The titles of the last two books do not begin with capital T.)

AoPS sells printed or online books or a combination of both. They also offer live, online options for all courses and a self-paced-online option for Prealgebra and Introductory Algebra A. (The online courses might be a great way for eager students to find the community support they need to enter competitions.)

The printed textbooks have separate solutions manuals with worked-out solutions for every problem. The online books include the solutions, and they also integrate the textbooks with interactions with the AoPS community, Alcumus (described below), and the free videos (also described below). The textbooks vary in length; those for the standard courses (except Calculus ) run from 528 to 720 pages, while other courses have from 256 to 400 pages.

Free videos are available online for Prealgebra, Introduction to Algebra , and Introduction to Counting & Probability . You can view these without having to pay or register. The videos do not replace the textbooks or online classes but supplement them. There are one or more videos for each lesson, all taught by Richard Rusczyk, a very engaging presenter as well as a former USA Mathematical Olympiad winner. I highly recommend watching them.

How the Courses Work

The courses divide the content into chapters, with several lessons within each chapter. Each lesson begins either with brief instruction or a set of three or more problems. Students should try to solve the problems on their own. The lesson continues with thorough explanations for how to solve each problem, and this is where most of the instruction is presented. This strategy very much reflects the title of the series, the Art of Problem Solving—students are focused on developing problem-solving skills as well as accuracy.

After studying the solutions, students have another set of problems to solve, a few of which are drawn from advanced math exams (no longer in use), such as the AHSME (American High School Mathematics Examination).

Lessons often use blue boxes to highlight key concepts, important ideas, and warnings about common mistakes.

There are Review Problems at the end of each chapter but no quizzes or tests for any of these courses. (The second diagnostic test for each course, titled “Do You Need This?,” could function as a final exam if needed.) The publisher’s explanation to me regarding this was: “Since our curriculum focuses on teaching students mathematical concepts and problem-solving skills, we believe that students who are using our textbooks have mastered the material if they can successfully solve the Review Problems at the end of each chapter.”

Alcumus, AoPS’s online learning system, is available for free to all students, even if they have not purchased any AoPS course. Alcumus adapts to the student’s performance, giving them problems to solve that are appropriate for their level—problems to solve in addition to those in their course. Alcumus can be used alongside the Introductory Curriculum courses, whether in print or online. (Students using courses from other publishers for pre-algebra through geometry should also find the program useful.) Alcumus provides ambitious students with work that will both reinforce and stretch their skills.

The AoPS website offers many other resources for advanced math students, including information about competitions, online forums, and training for competitions.

AoPS math courses should be fantastic for avid math students who are eager to learn and go deeper, but they also offer excellent and thorough instruction for the average student.

Pricing Information

When prices appear, please keep in mind that they are subject to change. Click on links where available to verify price accuracy.

See the publisher's website for options and prices.

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Publisher's Info

• Art of Problem Solving, Inc.
• PO Box 2185
• Alpine, CA 91903
• [email protected]
• https://artofproblemsolving.com/

Note: Publishers, authors, and service providers never pay to be reviewed. They do provide free review copies or online access to programs for review purposes.

Disclosure of Material Connection: Some of the links in the post above are "affiliate links." This means if you click on the link and purchase the item, I will receive an affiliate commission. Regardless, I only recommend products or services that I believe will add value to my readers. I am disclosing this in accordance with the Federal Trade Commission's 16 CFR, Part 255 "Guidelines Concerning the Use of Endorsements and Testimonials in Advertising."

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AoPS reviews?

Hello there fellow mathematicians. I´m thinking in buying the Art of Problem Solving books (the classic ones on contest preparation) but I wanted some reviews or advices from someone who already has used them. If anyone knows about the rest of the AoPS book series or the online courses it would be of great help too. Thanks a lot!

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Community Review for Art of Problem Solving

I wanted to clarify the misconceptions in several of the previous reviews. Art of Problem is first and foremost a textbook and online class company. The website is meant as an adjunct resource for these purposes. So while there are several online tools like Alcumus on the site. These were never meant to be stand-alone.

Secondly, the primary focus of all the textbooks is problem solving. AoPS offers an extremely rich/deep take on traditional topics. The problem organization in the books is particularly brilliant with problems building on each other towards a more complex view of each topic. The style of questions is structured around interesting samples from contest math with a focus on rigorous proofs and building techniques for attacking harder problems. I highly recommend reading a sample chapter of the books and judging the material for yourself.

The books take a very constructionist approach with the students being asked to first try to solve a group of exercises and then explanations are given and structure is built up. For a few of the books i..e Algebra/Pre-Algebra there are companion videos which I've found useful. The overall level is geared towards advanced students. In practice, many kids find the jump from other texts to AoPS to be difficult initially because of the added depth and rigor. I can't imagine as someone previously suggested using any of this as a remedial program.

Instead, either you should the website as an additional resource if you're using the AoPS textbooks or it also works well as an enrichment resource with another curriculum. I'd suggest interleaving some of the material as appropriate and particularly as challenge problems within the normal sequence.

How I Use It

Art of Problem Solving is great as a problem bank especially for interesting non-standardized problems. The books actually work best for this. You can take sections out and enhance many topics. It might be useful for advanced learners but only if they're fairly self-sufficient.

More community reviews for Art of Problem Solving

Great curriculum, no real world applications, pure math(this is good), strong program with emphasis on deep understanding of concepts and love of math. would highly recommend above most curriculums i have tried., two stars for remedial. five stars for advanced..

It's a new school year! Make it the best yet with math, language arts, and contest training courses!

Unlock Your Student's Future Potential

Advanced Online Math and Language Arts Courses for Grades 2–12

Starting at $50/week

Since 1993, Art of Problem Solving has helped train the next generation of intellectual leaders. Hundreds of thousands of our students have gone on to attend prestigious universities, win global math competitions, and achieve success in highly competitive careers.

Why Students & Parents  AoPS Academy

"My son LOVES his class! The math is inspiring, the teacher has a wonderful sense of humor, and he got to meet other students who love math as much as he does. Amazing job! Thank goodness we found AoPS!" — Excited Dad

"The staff at AoPS Academy do a great job helping students reach their fullest potential. My children have learned to think logically and creatively through solving tough problems. I'm glad we found this valuable resource!" — Usha D, Parent

Year-Round Math

Grades 2–12

Using our world-renowned AoPS and Beast Academy textbook series, each level provides a full year math curriculum that dives deep into all core topics

Honors Math 2

Honors math 3, honors math 4, honors math 5, honors math 6: prealgebra, contest math 6: prealgebra, honors math 7: introduction to algebra, contest math 7: introduction to algebra, honors math 7.5: counting, probability, and number theory, honors math 8: introduction to geometry, contest math 8: introduction to geometry, honors math 9: intermediate algebra, honors math 10: precalculus, honors calculus, high school contest math, year-round language arts.

Engaging in this full curriculum designed for motivated learners, students will learn the skills required to succeed in high school and beyond by building a strong foundation in all aspects of language arts

Honors Language Arts 3

Honors language arts 4, honors language arts 5, honors language arts 6: foundations in middle school language arts, honors language arts 7: persuasive writing and rhetoric, honors language arts 8: research, presentation, and public speaking, summer math.

Grades 3–12

Preparing students for the next school year and math contests like AMC and MATHCOUNTS through enriched study and exploration beyond the standard curriculum

Math Beasts Camp 3

Math beasts camp 4, math beasts camp 5, math beasts camp 6 (prealgebra prep), math beasts camp 7-8 (algebra prep), math beasts camp 8-9 (geometry prep), middle school math contests: number theory and geometry, middle school math contests: algebra and counting, amc 10/12 prep camp, summer language arts.

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Students put their creative and analytical skills to the test with Language Arts enrichment camps designed to enhance reading, writing, and persuasive speaking skills

Creative Writing

Writing for the spotlight, language arts triathlon, rhyme-bot derby: language craft and analysis.

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Mock trial: persuasive speaking, academic essay writing, go above and beyond with aops.

Solve tough problems through immersive active-learning experiences

Collaborate with exceptional classmates from across the country.

Participate in lively, interactive activities through a video conferencing format

Ready to start solving problems?

Art of Problem Solving, Algebra

If your student has completed a 6th grade math program and you’re looking for something more rigorous, the Art of Problem Solving Pre-Algebra and Introduction to Algebra could be a good fit. Be warned, though; AoPS unapologetically states: The text is written to challenge students at a much deeper level than a traditional middle school prealgebra course.

Developed by the mathematicians who created the Mandlebrot competition, AoPS takes math beyond the “plug and chug” method of memorizing formulas and then just crunching numbers with a calculator. Instead, students are taught how to solve problems – lots and lots of problems.

AoPS is extremely text-heavy, which could be a problem for kids who need visual supports to help them stay focused. Pages are packed with information, kind of like reading a word-for-word lecture that a teacher may give in class. The good part is that, if you can make it through all the reading, you probably won’t have many questions. It’s that thorough and well-explained.

Parents don’t have to be strong in math to be able to use this text with their kids – as long as you do the lessons together. The handy icon system helps you identify tricky concepts you might want to spend extra time covering. The Solutions Manual gives answers to every problem. It also includes an explanation for how to get the answer for many of the Review and Challenge problems.

Check out the Free Alcumus site , hosted by AoPS . Try some of the thousands of problems that supplement the Pre-algebra and Intro to Algebra text and you’ll get a feel for whether this program will be a good fit for your math loving kid.

Not sure what level you should start with? Visit the AoPS website’s Recommendation page and look for the Are You Ready? and Do You Need This? links to find placement tests for each textbook. If you’re still stuck, you can contact AoPS staff and they’ll help you decide.

Date of Review: 05/26/2015

Alessandra homeschooled her kids from kindergarten into college. In the early 2000s, she founded the Maryland Homeschool Association, a secular and inclusive statewide advocacy group. With a master;s degree in education, Alessa has written curriculum for nonprofis, a local community college, as well as supplemental material for homeschool families.

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Get it from Amazon for $9.95  (available in three colors).

16. A pack of Bottle Bright tablets that'll do all the hard work cleaning your gross bottles. No scrubbing needed!

Just fill your dirty mug with water, add a tablet, let fizz for half an hour, and then rinse!

Promising review: " I seriously can't believe how well these little tablets work. I had the most disgusting HydroFlask coffee mug — I tried everything. Bottle brush, different kinds of soap, vinegar, EVERYTHING, and there was still a seemingly impenetrable layer of black sludge inside. It grossed me out so much that I bought ANOTHER HydroFlask to replace it. Now, eventually, they both got black inside, so I tried these little magic tablets.  I let a tablet sit in each of my mugs for an hour or two, and the black sludge rinsed out COMPLETELY without even scrubbing. They look brand new. What a magical product. No weird smell or taste afterward, either. I liked it so much that I used it on another HydroFlask I use primarily to hold my Bloody Mary mix in the fridge. I couldn't put anything else in it because it would always taste like spicy tomato juice. Bottle Bright to the rescue! No residual smell or taste anymore. AMAZING AMAZING AMAZING." — Amazon Customer

Get a pack of 12 from Amazon for $7.99 .

Check out our closer look at these Bottle Bright tablets for more incredible before and afters.

17. A wildly popular hair serum packed with vitamin E, aloe vera, and argan oil to ensure your mane stays silky soft and frizz-free, no matter how humid it gets (Florida bbs, this one is for you).

Promising review : "I found this product through a BuzzFeed article for those with frizzy hair, and THIS PRODUCT DID NOT DISAPPOINT, PEOPLE! I put this serum through the ringer; I got this to stand up to Bay Area fog AND the nasty, humid heat wave we got a couple of weeks ago. No frizz, no extra curls. Nothing! On top of that, it made my hair super glossy. I'm very impressed. Heads-up, though, a little goes a long way. One pump is enough to cover my hair and I have medium-length hair. Also, it can make your hair look oily if you use too much. I do have oily hair, so working on my ends first, then working to my roots works for me." — Amazon Customer

Get it from Amazon for $9.99 .

To learn more, check out " Thousands Of People Say This Serum *Actually* Has Helped Their Dry And Frizzy Hair ."

18. Medicated, waterproof bandages that will make your wart erupt from the depths of your skin like a gross butterfly. The concentrated salicylic acid works hard to bring your skin back to its former wart-free glory.

Don't use these medicated pads on irritated, red, or infected skin. And only use these for warts — no ingrown hairs, genital warts, or moles.

Promising review: "I recently purchased this for my 9-year-old son’s wart, and I am beyond impressed with the results. The product worked quickly and effectively, removing his wart in just a couple of days. Compared to other brands and products I’ve tried in the past, Compound W is by far the most effective. I would purchase it again without hesitation and highly recommend it to anyone dealing with warts. It’s cheap, quick, and truly effective. Don’t hesitate to buy it — this product is a game-changer!" — Helen

Get a pack of 14 from Amazon for $6.96 .

19. A blessedly effective carpet cleaner compatible with most carpet cleaners and capable of turning your brown carpet into whatever color it was when you bought it.

Promising review: "This is a wonderful product. I have an older dog who can't always wait until I get home from work, so accidents occasionally happen. I have a Hoover Rug Scrubber , and this product is very easy to use with my machine. Carpet Miracle works amazingly well on stains and odors, even those that have set for a while. I did the carpet in the 'room of shame' twice just to be sure, although for normal soil and odors, once should be more than enough. It leaves behind a wonderful clean, fresh scent, and the carpet feels and looks wonderful. I plan on continuing to use this product to keep my carpets clean and my home smelling fresh." — Vicky Faurot

Get it from Amazon for $19.97 .

20. A bottle of sulfate-free biotin shampoo  that thousands of reviewers have said helped volumize their hair. The biotin can help fill out thinning hair and the nourishing ingredients like rosemary oil, zinc, and coconut oil can help moisturize. 

Read more about how biotin shampoo could help with the fullness of hair at Cleveland Clinic .

Maple Holistics is a small business that specializes in beauty products with all-natural ingredients. Check out a TikTok of the biotin shampoo in action. 

Promising review: "I love this shampoo! I had gastric sleeve surgery, and my hair was very, very thin. I started taking biotin and switched to biotin shampoo, as recommended by my doctor. It doesn’t make your hair grow by washing it, but it makes your hair and roots stronger. And it worked. After using the whole bottle, my hair was stronger. I had a lot of new growth that was sticking around (the previous new growth fell out early on), and my hair was getting thicker. Eight months later, my hair is back to normal. It's longer and thicker than ever."  — Diane J. Huff

Get it from Amazon for $12.95  (available in three scents).

21. A jewelry cleaning pen to bring that brilliant shine back to your cloudy gems, no costly trip to the jeweler required.

The pen is filled with a cleaning and polishing solution to remove clouding dirt and oil, leaving gems shiny. Just twist the bottom of the pen to dispense the fluid and brush on.

Promising review: "I worked several years in a fine jewelry store that sold this product. We used the Dazzle Stik from time to time to add a quick shine to merchandise that was just tried on and on days our jeweler was not there to professionally clean items. I was always impressed with the instant shine the diamond stick provided. The brush tip is very helpful for getting under the stones and in the setting where gunk builds up reducing that brilliant shine. I recently got engaged and immediately purchased a Diamond Dazzle Stik of my own. It's the perfect way to quickly give my ring some shine before leaving the house. You simply brush, rinse, dry and go. I don't feel this product replaces the need to professionally clean your jewelry every now and then, but it does help with the maintenance of that new shiny sparkle." — Amazon Customer

Get it from Amazon for $8.17 .

22. A hair-finishing stick that works like a plant oil-covered mascara wand to flatten those annoying flyaway hairs on the back of your neck or around your hairline.

Or for more hold, check out this effective edge control gel !

Promising review: "I have super fine hair with lots of breakage. No matter how I put my hair up I have bad flyaways. This stuff is great! So easy, and it stays better and looks smoother than stiff hair spray." — Jill Stilfield

Get it from Amazon for $7.98 .

23. Columbia hiking shoes  beloved by outdoorsy folks all over the country thanks to their unmatched comfort and style. Their high-traction rubber sole and waterproof leather-and-suede upper are designed to hold up during intense hikes in all kinds of weather. And they're pretty cute, as far as hiking boots go. 

Prime members: You can  try before you buy !

Reviewers swear these don't even have a break-in period , making them a great pick for that last-minute outdoorsy trip where hiking and rough terrain are on the docket. Note that  it's recommended to go up a half size in these boots to accommodate thicker hiking socks. 

Promising review : "There’s a reason that these hiking boots are the #1 best hiking boots on Amazon. I am amazed that they are so comfortable. This is day four that I’ve worn them, and my feet feel fantastic! I hiked four miles on the second day, and my feet were not tired, achy, or sore. I have been hiking for 30 years and have had numerous boots, but not a single pair has been this comfortable.  The price is so worth it for what you get." — Granny B

Get them from Amazon for  $49+  (available in sizes 5–12, including wide sizes, and 16 colors). You can find a similar men's version  here .

24. A set of OXO's amazing pop containers so you can keep your dry food and ingredients fresh and pest-free longer. They'll also make your pantry look a lot less chaotic.

These sleek-looking containers keep food fresh and presentable, so you don't have to get a lecture when your mom visits and peeks into your cabinets. They come in a variety of shapes and sizes to accommodate your various needs. So far, I've used them for sugar, flour, cornmeal, Wheat Thins, spaghetti, pretzels, and almonds. Each container has a big button on top that you can press to open or seal the jar. It's so satisfying to pop these things open that I can't imagine using a different container system ever again.

Get them from Amazon: the five-piece set for $49.95 and the 10-piece for $112.95 .

25. A leather cleaner to bring your furniture, car seats, bags, shoes, and more back to usable condition. The solution conditions, cleans, and restores — so maybe you don't need to buy a new couch after all.

Promising review: "I've used this on all of my favorite shoes and boots. Then I used it on my purse. It's enjoyable seeing how much improvement you can see on beat-up leather. The scent is what sold me on this product. I loved that it was an almond scent. But the results are so much better. It took about two minutes to clean them up. Highly recommended. Also, their customer service is top-notch." — RileyD

Get it from Amazon for $34.15 .

26. A tote organizer that has two vital functions: 1. brings order to your chaotic bag that weirdly has no pockets, and 2. lets you transport all your stuff with one fluid motion. It adds a whopping 13 pockets and a detachable middle pouch for valuables.

Measure your bag to make sure it's a good fit!

Promising review: "This thing is great! I have a large bag I use as a carry-on when I fly by plane. Unfortunately, it has very few pockets and is just like a bottomless pit. This fit in perfectly. I was able to organize all my items quite well. They were easy to retrieve in flight and in the airport. I didn't use the zipper compartment at all but put that piece in to use it as a divider. It is well-made and sturdy. Be sure to measure your inner bag space first!" — Susan

Get it from Amazon for $19.88+ (available in five sizes and 13 colors).

27. A  shoe stretch spray  if you have a pair of gorgeous loafers or boots you love but never wear...because of the agonizing pain that comes with it. This spray will gradually stretch them out without staining or fading and is likely more effective than the freezer trick. 

BuzzFeed Shopping editor  Amanda Davis  swears by this spray. Check out her Foot Matters Shoe Stretch spray review  for more details! BTW, Foot Matters is a small business that specializes in shoe-stretching products.  

Promising reviews: "My sandals were awfully tight on the top of my feet. I tried everything before finally finding this spray. Nothing helped; there was no relief. Then I tried this spray, and it moved heaven and earth, and now my sandals are perfect! " — Swissharpist

"I injured my foot a few months back, and most of my shoes don't fit. I've tried stretchers, hair dryer, wearing multiple socks — everything. I bought this stuff, and within a half hour, my problems were solved . Try this method first." — Susan Olson

Get it from Amazon for $9.99 . 

28. A privacy window film that's easy to install and fills your home with lots of rainbows when the sun hits it just right. Now you can sing "somewheeeere over the rainbow" as you step over your dog.

Read more about this beloved rainbow window film .

Promising review: "I initially purchased this window film to use to brighten up my classroom. That film has lasted over two years! I enjoyed it so much that I bought another roll to use in my home. I would recommend using a spray bottle and a credit card-like object for application. Go slowly, one area at a time, to minimize bubbles as much as possible. My classroom ones have quite a few bubbles (user error), but that has not caused them to fall off, which has been very nice." — Kris H.

Get it from Amazon for $13.98+ (available in nine sizes).

29. A foaming lavender bath with pure epsom salt to soothe the body and create the most sudsy bath experience ever.

While you're in that extra deep bath, you might as well go all out and add some bubbles.

Promising review: "I love this stuff soooo much. I have had some rash issues on my fingers ever since I got a full-body rash from Downy Unstopables and had to take prednisone to get rid of it. Ever since I started regularly soaking in the tub with this, I have had hardly any flare-ups. Something about the ingredients in this particular combination makes my skin sing. Plus, it smells amazing and provides top-notch bubbles. Worth every penny. "— Sarah

Get it from Amazon for $6.69 .

30. A c ruelty-free Essence Lash Princess lengthening mascara that's lightweight, won't transfer, doesn't smudge, and lasts all day — oh, and makes your eyelashes look like tall, beautiful skyscrapers.

Promising review: "Love this mascara! I’ve used the 'best of the best' designer, super hyped, ultra-popular mascara available, and NOTHING compares to this stuff ! I don’t even bother using a lash curler anymore! I buy two at a time so I always have a spare (yes, it’s that good!!). I highly recommend this particular mascara... it goes on effortlessly, it lasts pretty much all day, and the price — I feel like I’m almost stealing it! I’d give it 10 stars if I could!!! " — chulaboola

Get it from Amazon for $4.99 .

31. A container of oatmeal paw butter to soothe and relieve your dog's cracked, dry paws. Its eco-friendly formula is pH-balanced so it's safe for cats, even if licked or swallowed. *seriously considers licking it myself to see if it really tastes like oatmeal*

Promising review:  "My dog's paws are extra soft, bringing back memories of when he was a puppy. My dog did not lick it, but I’m sure some dogs will. I recommend putting it on at night, so the oils do not get on your floors. In the morning, it is dry, and you we love to touch your dog's paws." — Joshua Torres

Get it from Amazon for $7.59+  (available in two sizes and scents).

32. A robot vacuum happy to do all the hard work for you. Just turn it on and watch someone else vacuum for a change. This device earned over 9,000 5-star reviews by being affordable, powerful, and able to reach tight corners.

This device is quiet, self-recharging, uses an infared-sensor to expertly maneuver around obstacles. And it comes with a 12-month warranty in case anything happens.

Promising review: "After running Eufy for nearly two months now, I can confidently say I absolutely LOVE it! It is amazing on my hardwood floors. It picks up sooo much fine dust brought in by my two German shepherds! They shed year-round! And it’s super effective with pet hair! Wow! I use Eufy once a day for a little more than hour, emptying and cleaning the brushes out about three to four times during use. It seriously has been life-changing, keeping my floors cleaner than sweeping! Great purchase and I would highly recommend!!" — Jacqueline M

Get it from Amazon for $249.99+  (available in three versions).

33. A box of Color Catcher sheets to prevent the dreaded "all my clothes are now pink from this one stray red sock" debacle. These handy sheets are designed to grab any dye that leaks or bleeds so you can keep enjoying your white tees.

Promising review: "These Color Catcher sheets really protect colors from bleeding. It makes all clothes/laundry brighter and looking like new — not that old washed color after just one wash without these sheets. I give this product 5+ stars. Highly recommend. A few pennies spent additionally on your laundry makes one look like wearing new clothes all the time." — Nikita

Get 72 sheets from Amazon for $10.95 .

34. A hairbrush cleaning tool  perfectly designed to remove hair, debris, fuzz, fur, dandruff — I could go on. OK, you see how this thing is very needed, right? 

The pointed end of the tool is ideal for picking out tangles, the stiff bristles work great at removing dust and dirt, and the bristles at the bottom end of the tool can give your hairbrush a deep clean.

Promising review: "When I bought this, I was really skeptical. I am grossed out by my hairbrush and clean it all the time. I've soaked it in vinegar and tea tree oil, used a toothbrush, scrubbed it with shampoo, picked it clean by hand — everything you can think of. Still, it's so hard to get the little lint that forms at the base of the brush — they don't dissolve off, and you have to pick them off the bristles one by one because a toothbrush won't even break them up. This thing works amazingly. I don't know why, it just does — the particular texture of the bristles on this really grabs everything. It scraped the little lint rings right off, and now my brush is cleaner, 10 times as fast." — HeartsofHavoc

Get it from Amazon for $11.95 .

35. A  multi-purpose cleaning paste to help you get your oven into tip-top shape in no time at all. It's made in small batches with natural ingredients, so it's safe for your family, pets, and home. Use it on your oven, stovetop, bathroom tiles — or even your produce and hands. 

BuzzFeed Shopping editor  Danielle Healy swears by this cleaning paste: "I recently moved into a new apartment, so lately I've been cleaning A LOT. Shortly after moving in, my partner and I went to store something on the top of the kitchen cabinets, only to find them caked in a thick layer of grime. 🤢 After going at it with regular multipurpose cleaner (with little success), we broke out the scour paste as a last-ditch effort, and OH BOY did this miracle product deliver (pics above). If it can handle that grossness, it's going to have no problem with day-to-day messes like soap scum and burnt-on food. Plus, it smells delightful and comes in minimal, low-waste packaging!"

Promising review : "Buyer beware: don’t buy just one!! Learn from my mistake, and buy at least two jars because this stuff is INCREDIBLE. I used it on my textured shower floor. It was EASIER to clean and required less scrubbing than any other product I’ve used , including CLR. If you have black matte bathroom fixtures, this is the stuff you need. My shower and sink drains look brand new. My other favorite part about this scour paste is the fact that I don’t need to use any paper towels. Just a wet rag to wipe it all down at the end. I will be buying more jars for myself and to give to my friends because this stuff is absolutely amazing. You won’t regret it!" — littleashleyshortcak

Get it from Humble Suds on Etsy for $15.95+ (available in two sizes/packaging materials). 

Reviews have been edited for length and/or clarity.

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Improved multi-strategy adaptive Grey Wolf Optimization for practical engineering applications and high-dimensional problem solving

• Open access
• Published: 05 September 2024
• Volume 57 , article number  277 , ( 2024 )

Cite this article

You have full access to this open access article

• Mingyang Yu 1 ,
• Jing Xu 1 ,
• Weiyun Liang 1 ,
• Sixu Bao 1 &
• Lin Tang 2  

The Grey Wolf Optimization (GWO) is a highly effective meta-heuristic algorithm leveraging swarm intelligence to tackle real-world optimization problems. However, when confronted with large-scale problems, GWO encounters hurdles in convergence speed and problem-solving capabilities. To address this, we propose an Improved Adaptive Grey Wolf Optimization (IAGWO), which significantly enhances exploration of the search space through refined search mechanisms and adaptive strategy. Primarily, we introduce the incorporation of velocity and the Inverse Multiquadratic Function (IMF) into the search mechanism. This integration not only accelerates convergence speed but also maintains accuracy. Secondly, we implement an adaptive strategy for population updates, enhancing the algorithm's search and optimization capabilities dynamically. The efficacy of our proposed IAGWO is demonstrated through comparative experiments conducted on benchmark test sets, including CEC 2017, CEC 2020, CEC 2022, and CEC 2013 large-scale global optimization suites. At CEC2017, CEC 2020 (10/20 dimensions), CEC 2022 (10/20 dimensions), and CEC 2013, respectively, it outperformed other comparative algorithms by 88.2%, 91.5%, 85.4%, 96.2%, 97.4%, and 97.2%. Results affirm that our algorithm surpasses state-of-the-art approaches in addressing large-scale problems. Moreover, we showcase the broad application potential of the algorithm by successfully solving 19 real-world engineering challenges.

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• Artificial Intelligence

Avoid common mistakes on your manuscript.

1 Introduction

The rapid advancement of science, technology, and industry has given rise to a multitude of intricate optimization problems. These problems frequently entail numerous variables, constraints, and objectives. Their solution spaces are huge and complex, and it is difficult for traditional deterministic optimization methods to obtain satisfactory solutions in acceptable time (Deng et al. 2022 ; Guo et al. 2023 ; Zhou et al. 2022 ). To cope with these challenges, researchers in the field of computational intelligence have started to search for new approaches. Among them, metaheuristic algorithms have attracted much attention due to their high efficiency, universal applicability and powerful global search capability (Aldosari et al. 2022 ; Chauhan et al., 2024; Chen et al. 2023 ).

When dealing with engineering problems, constraints are a crucial consideration. Constraints may be physical limitations or requirements and restrictions of the project. In the field of engineering, there are various techniques available for handling these constraints to ensure that projects proceed as expected and achieve their intended goals. One common technique for constraint handling is optimization algorithms. Optimization algorithms assist engineers in finding the best solution given certain constraints. These algorithms can be mathematical optimization methods such as linear programming, integer programming, or nonlinear programming, or they can be heuristic algorithms such as genetic algorithms, simulated annealing, or particle swarm optimization. By leveraging these algorithms, engineers can find the optimal design or decision solution while taking into account various constraints (Fu et al. 2024b ; Li et al. 2023 ).

MH algorithms are inspired by certain phenomena in nature, such as PSO (Kennedy and Eberhart 1995b ), Firefly Algorithm (FA) (Yang 2009 ), Sine Cosine Algorithm (SCA) (Mirjalili 2016 ), Wind Driven Optimization (WDO) (Bayraktar et al. 2010), Fruit Fly Optimization Algorithm (FOA) (Pan 2012 ), Competitive Swarm Optimizer (Chauhan et al. 2024 ), Fox optimizer(FOX) (Mohammed and Rashid 2023 ), Fitness Dependent Optimizer(FDO) (Abdullah and Ahmed 2019 ) and so on. These algorithms often do not rely on the specific nature of the problem, but instead draw on nature's strategies for stochastic search, which can effectively avoid falling into local optima (Abdel-Basset et al. 2023 ). With the development of deep learning, neural networks and other machine learning techniques, researchers have begun to try to combine these techniques with metaheuristic algorithms to further improve the efficiency of solving complex optimization problems (Garg et al. 2023 ). In recent years, with the wide application of heuristic intelligent optimization algorithms in numerical optimization solving, various swarm intelligence algorithms have been proposed (Fu et al. 2022 ).

The popularity of MH algorithms has four distinct advantages: practicality, generalizability, non-leading properties and avoidance of local optima (Fu et al. 2023a ). First, based on their natural theoretical framework, these strategies are relatively intuitive to construct and deploy, thus allowing engineers and researchers to rapidly integrate them into concrete applications (Havaei & Sandidzadeh 2023 ). Next, since these algorithms treat the problem as an unknown mechanism, they can be applied to a wide range of different tasks such as selection (Said, Elarbi, Bechikh, Coello Coello, & Said, 2023), shop visit balancing (Xia et al. 2023 ) and engineering problems (Nadimi-Shahraki et al. 2022 ). Further, these methods do not rely on derivative information and are particularly good for dealing with nonlinear problems (Aldosari et al. 2022 ). Ultimately, with the help of a global search strategy and a stochastic strategy for updating the location, they can efficiently jump out of the local optimum, which is particularly effective for those scenarios where there are multiple locally optimal solutions.

The existing MH algorithm mainly include: Physics-based algorithms (PhA), Swarm Intelligence Algorithms (SI), Natural Evolutionary Algorithms (EA), and Human-based algorithms (Abualigah et al. 2021 ). In the course of evolution, cooperative behavior between individuals has been gradually formed through natural selection over a long period of time. For example, Trojovský and Dehghani, 2022 proposed a pelican optimization algorithm (POA) inspired by pelican predation. The Genetic Algorithm (GA) is a method of optimization based on the principles of natural evolution. It was proposed by John Holland and his colleagues. is a typical example inspired by Darwinian evolution (Bäck & Schwefel 1993 ). Differential Evolution (DE), based on the concepts of natural selection and reproduction in Darwinian evolution(Storn and Price 1997 ); Genetic Programming (GP), inspired by biological evolution processes; and Evolution Strategies (ES) (Wei 2012 ). Among these, Genetic Algorithms (GA) and Differential Evolution (DE) are widely considered to be the most popular evolutionary algorithms, having garnered significant attention and being applied in numerous applications. Physical method is the result of the interaction of physical law and chemical variation. For example, the Chernobyl Disaster Optimizer (CDO) is an optimization algorithm inspired by the core explosion at the Chernobyl nuclear power plant. (Shehadeh 2023 ). The Galaxy Swarm Optimization (GSO) (Muthiah-Nakarajan and Noel 2016 ) algorithm, inspired by the motion of galaxies; the Firefly Algorithm (FFA), drawing inspiration from soil fertility in agriculture (Shayanfar and Gharehchopogh 2018 ); the Firefly Algorithm (FFA), drawing inspiration from soil fertility in agriculture (Eskandar et al. 2012 ); and the Gravitational Search Algorithm (GSA), derived from Newton's law of universal gravitation and kinematic laws (Rashedi et al. 2009 ).In sharp contrast, a variety of human behaviors are simulated based on human behavior patterns, such as the "Alpine skiing Optimization (ASO)" proposed by Professor Yuan, a new idea influenced by the competitive behavior of athletes. Each of these different Metaheuristic algorithms has its own characteristics. According to different problems and requirements, appropriate algorithms can be selected to solve the optimal problems (Yuan et al. 2022 ). Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995a ) is inspired by the foraging behavior of bird flocks and fish schools. The Ant Colony Optimization (ACO) algorithm (Dorigo et al. 2006 ) is inspired by the social behavior of ant colonies during foraging. The Pathfinder Algorithm (PFA) (Yapici & Cetinkaya 2019 ) is inspired by the collective action of animal populations in finding optimal food areas or prey. The Harris Hawk Optimization algorithm (HHO)(Heidari et al. 2019 ) is based on the predatory process of Harris hawks hunting rabbits. The Sparrow Search Algorithm (SSA) (Xue and Shen 2020 ) is inspired by the foraging and anti-predatory behavior of sparrows. The Dung Beetle Optimization algorithm (DBO) (Xue and Shen 2022 ) is inspired by the rolling, dancing, foraging, stealing, and reproductive behaviors of dung beetles. The Remora Optimization Algorithm (ROA) (Jia et al. 2021 ) is inspired by the behavior of remoras adhering to different-sized hosts to facilitate foraging. The Black Widow Optimization algorithm (BWO) (Hayyolalam and Kazem 2020 ) is inspired by the unique reproductive behavior of black widow spiders. Dikshit Chauhan et al. proposed the Artificial Electric Field Algorithm (AFFEA) based on a series of learning strategies (Chauhan and Yadav 2024a , b ). Additionally, the Secretary Bird Optimization Algorithm (SBOA) was introduced based on the survival behavior of secretary birds in their natural environment (Fu et al. 2024b ), while the Red-Billed Blue Magpie Optimizer (RBMO) was proposed by simulating the search, chase, prey attack, and food storage behaviors of red-billed blue magpies (Fu et al. 2024a ).

Generally, the optimization process of the MH algorithm can be divided into two main steps (Saka et al. 2016 ): exploration and exploitation. In the exploration phase, the algorithm mainly focuses on searching all corners of the solution space to ensure that no possible optimal solution area is missed; while in the exploitation phase, the algorithm will focus on known high-quality solutions and further deepen the search in order to find the real the optimal solution. These two phases complement each other and ensure that the algorithm has both breadth and depth. GWO is inspired by the hunting behavior of grey wolves (Mirjalili et al. 2014 ). GWO effectively balances the two stages of exploration and exploitation by combining the social behavior of grey wolves with a dynamically adjusted location update strategy, thereby ensuring good global and local search capabilities.

Since its introduction in 2014, GWO has received widespread attention from scholars at home and abroad for its simplicity and efficiency, and has become an important tool for solving complex optimization problems (Fan and Yu 2022 ). However, similar to other optimization algorithms, the GWO algorithm does have some limitations although it has shown quite good performance in many optimization problems. In particularly, it is prone to suffer from prematurity and local optimality when dealing with multimodal function problems. As the iterative process of the GWO progresses, the inherent social hierarchy mechanism within the wolf population leads to a decrease in diversity. This mechanism prioritizes the positions and decisions of the leading wolves (Alpha, Beta, and Delta), influencing the entire pack's movement. As a result, the population tends to converge towards the leaders’ positions. However, this strong convergence driven by the social hierarchy can also lead to a drawback. The population may start to aggregate too closely or blindly around the leaders’ current positions. This phenomenon, often referred to as premature convergence, limits the algorithm’s ability to thoroughly explore the solution space. Consequently, the algorithm might struggle to escape local optima, as the current best solutions (guided by the leading wolves) might not always represent the global optimum. The pack, following the leaders too closely, can get trapped in these local optima, lacking the diversity or exploratory behavior needed to venture out and discover better solutions elsewhere in the search space. (Wang et al. 2018 ). In addition, when global exploration transitions to local mining, the algorithm may lose the ability to explore a wider solution space and overly concentrate on a specific region for detailed search. Such a centralized strategy, although helpful in finding the optimal solution in the local region accurately, may also lead the algorithm to ignore other promising regions (Wolpert and Macready 1997 ). Despite there has been various GWO variants, such as Advanced Grey Wolf Optimizer (AGWO) (Meng et al. 2021 ), Exponential Neighborhood Grey Wolf Optimization (EN-GWO) (Mohakud and Dash 2022 ), Hybrid Grey Wolf Optimizer with Mutation Operator (DE-GWO) (Gupta and Deep 2017 ), and others (Ambika et al. 2022; Biabani et al. 2022 ). These improved versions do not break through in solving the large-scale global optimization problems of CEC 2022 and CEC 2013. Moreover, their performance in dealing with complex problems remains unsatisfactory.

To improve the performance of the GWO, this study incorporates several key enhancements. Firstly, the search mechanism from PSO is employed to increase population diversity. This addition helps in broadening the search scope of the algorithm. Secondly, the IMF is used to adjust inertia weights, a strategy that aids in fine-tuning the balance between exploration and exploitation. Lastly, an adaptive mechanism based on the Sigmoid function is introduced for updating the positions of individuals within the population. This adaptive update strategy strengthens the group's ability to escape local optima, enhancing the overall effectiveness of the GWO algorithm in finding optimal solutions.

An improved adaptive grey wolf optimization (IAGWO) is proposed to address the shortcomings of the GWO algorithm. The main contributions are as follows.

The PSO search mechanism is introduced to enhances the algorithm's search efficiency and robustness by updating grey wolf positions early in each iteration. Additionally, the dynamic adjustment of inertia weights through the IMF boosts global search capability initially and local search effectiveness later.

Adaptive position updating strategy based on Sigmoid function to balance the exploration and exploitation of IAGWO.

To evaluate the exploration and exploitation capabilities of IAGWO, extensive experimentation is conducted using a suite of 67 test functions, which includes benchmarks from the CEC 2014, CEC 2017, CEC 2020, CEC 2022, and CEC 2013 for large-scale global optimization problems.

The effectiveness and accuracy of IAGWO in solving practical engineering design challenges are thoroughly assessed through its application to 19 diverse engineering design challenges.

The paper is organized as follows: Sect.  2 provides a brief review of the previous enhancements and potential application directions of the GWO. Section  3 details the original GWO algorithm and the proposed improvement strategy. Section  4 evaluates IAGWO performance through relevant experiments and in-depth analysis. Finally, Sect.  5 concludes this paper with a summary of the results and an outlook on future research directions.

2 Related work

In recent years, there has been a significant focus among researchers on enhancing the GWO. These improvements are aimed at boosting the algorithm's search performance and effectiveness. Scholars have explored various approaches to achieve this, including aspects such as adjusting the algorithm parameters, improving the speed and position equations, and combining it with other algorithms.

Yu et al. ( 2023 ) adopted a new update search mechanism, improved control parameters, mutation driven strategy and greedy selection strategy to improve GWO in the search process. (Singh and Bansal 2022a ) proposed a hybrid GWO and Differential Evolution (HGWODE) algorithm and applied it to UAV path planning. (Cuong-Le et al. 2022 ) introduced an equation to control the moving strategy of the algorithm in each iteration and proposed New Balance Grey Wolf Optimizer (NB-GWO), which was used to optimize the hyperparameters of the deep neural network for damage detection of two-dimensional concrete frames. Liu et al. ( 2023 ) proposed a hybrid differential evolution GWO (DE-GWO) algorithm and applied it to gas emission identification and localization. Luo et al. ( 2023 ) introduced butterfly optimization algorithm and opposition-based learning method based on elite strategy, adaptive nonlinear inertia weight strategy and random walk law to improve the shortcomings of slow convergence speed and low accuracy of GWO algorithm when dealing with high-dimensional complex problem. To address the issue of premature convergence encountered by the classic GWO in some situations due to the stagnation of sub-optimal solutions, Gupta et al. introduced an enhanced leadership-inspired grey wolf optimizer for global optimization problems (GLF-GWO)(Gupta and Deep 2020 ), Addressing the issues of slow convergence speed and insufficient global exploration in GWO, which can lead to settling in local optimal states and failure to achieve global optimal solutions, Singh et al. proposed a novel mutation-driven modified grey wolf optimizer (MDM-GWO) (Singh and Bansal 2022b ). MDM-GWO integrates new update search mechanisms, modified control parameters, mutation-driven schemes, and greedy selection methods into the search process of GWO. Addressing the issues of slow convergence speed and susceptibility to local optima in the Grey Wolf Optimizer (GWO) algorithm, Zhang et al. proposed a nonlinear control parameter strategy based on a sinusoidal function (GWO-SIN) and a nonlinear control parameter combination strategy (GWO-COM) (Zhang et al. 2019 ).

Soliman et al. ( 2022 ) proposed a novel hybrid African vultures–grey wolf optimizer (AV–GWO) approach to precisely estimate the electrical parameters of such TDM. Nadimi-Shahraki et al. ( 2021 ) introduced an enhanced variant of the Grey Wolf Optimization algorithm, termed I-GWO. The algorithm, based on a dimensionally learned hunting and searching (DLH) strategy, uniquely constructs hunting domains for each Wolf and enables them to share information about neighboring domains with each other. This enhances the algorithm's local and global search capabilities for more balanced performance, while also helping to maintain population diversity. A. Abushawish and A. Jarndal (Abushawish and Jarndal 2021 ) jointly proposed a new hybrid algorithm named GWO-CS that combines the advantages of Cuckoo Search (CS) algorithm and GWO algorithm. This algorithm primarily incorporates the position update equation from the CS to further refine the global search process of the GWO. Aimed at solving the problems of poor stability and easily falling into the local optimal solution in the grey wolf optimizer (GWO) algorithm, Liu et al. proposed an improved GWO algorithm based on the differential evolution (DE) algorithm and the OTSU algorithm is proposed (DE-OTSU-GWO) (Liu, Sun, Yu, Wang, & Zhou, 2020). The multithreshold OTSU, Tsallis entropy, and DE algorithm are combined with the GWO algorithm. The multithreshold OTSU algorithm is used to calculate the fitness of the initial population. The population is updated using the GWO algorithm and the DE algorithm through the Tsallis entropy algorithm for crossover steps. Multithreshold OTSU calculates the fitness in the initial population and makes the initial stage basically stable. Tsallis entropy calculates the fitness quickly. The DE algorithm can solve the local optimal solution of GWO. To address the issues of GWO's susceptibility to local optima and its low exploration capabilities, Hardi Mohammed et al. proposed the Enhanced GWO (EGWO) (Mohammed et al. 2024 ). EGWO employs diverse methods to improve the performance of GWO, utilizing gamma, z-position, and the golden ratio.

Liu et al. ( 2022 ) introduced a novel improvement strategy for the GWO algorithm, known as the exponential convergence factor improvement strategy. This strategy is designed to more accurately simulate the actual search process of grey wolves. It incorporates dynamic weighting factors and enhances control parameters to reduce the likelihood of the GWO algorithm getting stuck in local optima. However, despite these improvements, experimental findings indicate that GWO still faces challenges in accurately handling high-dimensional functions. Şenel et al. ( 2019 ) integrated a differential disturbance operator into the GWO algorithm. This addition brought an element of exploration into the exploitation phase, thereby enhancing the GWO algorithm's overall optimization capabilities. Jangir and Jangir ( 2018 ) proposed a multi-objective version of the GWO algorithm, named NSGWO. This algorithm utilizes a crowding distance mechanism to select the optimal solution from a set of Pareto optimal solutions. This approach helps guide the search towards the dominant region in multi-objective search spaces. NSGWO was tested on a variety of standard unconstrained, constrained, and engineering design challenges, demonstrating its efficiency and effectiveness in diverse optimization scenarios.

3 Methodology Overview: Standardized GWO and Proposed Enhancements

This section offers an overview of the hunting behavior and the mathematical model that forms the foundation of the original GWO. Additionally, we introduce the IAGWO, our proposed enhancement to GWO. IAGWO integrates the PSO search mechanism, the IMF strategy for inertia weighting, and an adaptive strategy for updating positions. These additions aim to refine and boost the efficiency of the original GWO algorithm.

3.1 The standardized GWO

3.1.1 inspiration of grey wolf packs’ hunting activity behavior.

The GWO algorithm draws inspiration from the hunting behavior of grey wolf packs. It mathematically simulates the way a group of grey wolves hunts, encircles, and targets their prey while adhering to a well-defined social hierarchy. In this hierarchy, the pack is led by three primary wolves: the Alphas ( α ), Betas ( β ), and Deltas ( δ ), each playing a crucial role in guiding the pack's movements and decisions. These wolves are considered the leaders, showcasing significant leadership abilities. Below them are the Omega ( w ) wolves, who occupy a subordinate role and follow the directives of the leading wolves. This hierarchical structure, integral to the functioning of the GWO algorithm, is depicted in Fig.  1 .

Hierarchy of the grey Wolf Pack

3.1.2 Mathematical model: GWO

GWO simulates grey wolf leadership and hunting mechanisms by dividing grey wolves based on their characteristics into a leader, α , who rules over the entire grey wolf; a facilitator, β , who helps α to make decisions and replaces α when α dies; and an enforcer, δ , who follows α 's and β 's orders (Fan and Yu 2022 ). GWO searches for excellence by modeling the wolf hunting process. In addition to the social hierarchy of wolves, group hunting is another interesting social behavior of grey wolves. The main phases of grey wolf hunting are as follows: The Grey Wolf Optimizer algorithm mimics the hunting behavior of grey wolf packs. Initially, in the "tracking, chasing, and approaching prey" phase, each wolf searches for potential solutions in the solution space and adjusts its position through certain search strategies to get closer to possible candidate solutions. Subsequently, in the "chasing, surrounding, and harassing prey until it stops moving" phase, the wolf pack collaborates to try to corner the prey into a smaller area and prevent its escape, involving behaviors such as encircling and harassing the prey to prevent its escape. Finally, in the "attacking prey" phase, once the prey is cornered and unable to escape, the wolves concentrate their attack on the prey, gradually optimizing the position of candidate solutions through strategies such as linear or leap searches until finding the optimal solution or meeting specific optimization criteria. These three phases represent the Grey Wolf Optimization algorithm's process of searching, chasing, and optimizing in the solution space, analogous to the behavior of a grey wolf pack during hunting, progressing from search to attack, gradually optimizing and approaching the optimal solution.

Now, this paper shows the calculation steps of the basic grey Wolf Optimization algorithm and the pseudo-code as follows (Algorithm 1). The GWO algorithm process is as follows:

1) Each member is initialized using Eq. ( 1 ), determine the population size N , the maximum number of iterations M , the single grey wolf dimension dim , and ɑ , A and C ;

where, LB and UB are the lower and upper boundaries of the solution space, respectively. X represent the positions of the current solution. phi is a random number between [0,1].

2) Calculate the fitness value of each individual using the test function. Then, based on the magnitude of the fitness values, select the best-fit individual as the α -wolf, the second-best individual as the β -wolf, and the third-best individual as the δ -wolf;

3) The mathematical model of Wolf pack leader tracking prey is shown in Eq. ( 2 ), which calculates the traction direction of the entire pack according to the distance difference between the Wolf leader and the pack, that is, the movement direction information of the pack, can be calculated as shown in Eqs. ( 3 ) and ( 4 ). Update the current grey wolf position according to Eqs. ( 2 )– ( 4 ).

where D α , D β , and D δ denote the distance difference between α -wolf, β -wolf, and δ -wolf and other individuals, respectively. X α , X β , and X δ indicate the current positions of α -wolf, β -wolf, and δ -wolf respectively, X indicate the current positions C 1 , C 2 , and C 3 satisfy the constraints of Eq. ( 6 ). A 1 , A 2 , and A 3 are random vectors satisfying the constrain of Eq. ( 5 ); X 1 , X 2 and X 3 are the traction directions of the three leading wolves; and X ( t  + 1) represents the next collective movement position of the wolf pack. As shown in Fig.  2 , the final orientation of the wolves in the search space will be randomly positioned within a circle defined by the locations of the α , β , and δ in the search space. This graphical representation illustrates how the wolves’ positions influence the movement and direction of the entire pack in the pursuit of their prey.

Position update of wolf groups in GWO algorithm

4) Update ɑ , A and C according to Eqs. ( 5 )–( 7 );

where, the parameter ɑ plays a crucial role in balancing global search and local exploration. Its value is set to decrease linearly from 2 to 0 over the course of the algorithm's iterations. Initially, a higher value of ɑ aids in the global convergence of the algorithm, guiding the wolf pack swiftly towards the region where the optimal solution might be found. As the algorithm progresses through its later iterations, the gradual decrease in the value of ɑ facilitates more refined exploration in the area of the optimal solution. This helps improve the convergence accuracy of the GWO algorithm, ensuring a more precise final result. r 1 and r 2 are random vectors and r 1 , r 2  ∈ [0, 1].

5) Update the positions of other individuals, calculate the updated fitness value based on the new position, and update the α -wolf, β -wolf, δ -wolf and global optimal solution, \(R\) represents the position vector of the optimization target;

6) Judge whether the specified stopping condition is reached (e.g., the maximum number of iterations is reached), if not, repeat steps 2 to 5. Otherwise, output the optimal result: the position of the α -wolf obtained at the end is the optimal solution, and the corresponding fitness value is the degree of superiority or inferiority of the optimal solution.

3.2 Improved grey wolf optimization algorithm

3.2.1 pso search mechanism.

The GWO (Grey Wolf Optimizer) exhibits a weak exploratory capability in its early stages and lacks diversity within the population, consequently resulting in suboptimal solution quality. In order to enhance exploration capabilities, improve population diversity (Hu et al. 2022 ), and increase the quality of solutions (Hakli and Kiran 2020 ), this study integrates the PSO, introducing a velocity concept to provide a new search mechanism for the GWO. The individual grey wolves are updated in terms of position during the early iterations, and the application in the velocity update introduces additional randomness. This prevents the algorithm from converging prematurely and encourages exploration of new areas, thereby increasing population diversity. By dynamically adjusting the velocity and position of each individual, this method may help in more effectively balancing global exploration and local exploitation. Leading to a wider search in the early stage of the iteration, this assists in identifying potential high-quality solutions. The computation Eq. ( 8 ) is as follows:

where t represents the current number of iterations, X and \({X}_{best}\) represent the positions of the current solution and the best-performing solution, respectively. \({v}_{rand}\left(t\right)\) is the velocity vector of the current solution at time t of iteration. phi is a random number between [0,1]. \({X}_{selfbest}\) is the best position vector in the history of the current solution.

In this study, at the start of each iteration, a PSO updating strategy is employed, along with the addition of extra randomness to stimulate a more extensive global search. This approach helps avoid local optimization and increases population diversity. This approach not only accumulates a more diverse and high-quality search experience for the GWO but also more effectively balances global exploration and local exploitation by dynamically adjusting the search behavior.

3.2.2 IMF inertia weighting strategy

Inverse Multiquadric Function is a decreasing function based on the principle of inverse multiple squares. It is often used as a regularization method in neural networks, such as a kernel function in support vector machines (Hu et al. 1998 ; Rathan et al. 2023). In accordance with the characteristics of the IMF, this paper incorporates it into the population position update mechanism within the framework of the GWO as delineated in Eq. ( 3 ). The IMF inertia weight ω , along with the revised formulae for the wolf pack updating process, are elucidated in Eqs. ( 9 )–( 10 ).

where, the parameter groups [a, b, c, d] are taken as [0.6, 0.02, 0.05, 0.3] and the graph of ω is shown in Fig.  3 . As indicated by Fig.  3 , during the early to mid-phases of the algorithm's iteration, the inertia weight ω is set to a higher value. This larger influence of the α -wolf, β -wolf, and δ -wolf on the updated positions is beneficial for the pack to quickly converge towards the optimal solution, effectively preventing the waste of search resources due to blind searching and thus enhancing the quality of the pack. As the development progresses to the mid and late stages and the pack becomes densely concentrated, if the higher-ranking wolves get trapped in a local optimum, the lower-ranking wolves led by them are also unable to escape this local optimum. At this juncture, the value of ω should be reduced to a lower level, thereby enlarging the pack's autonomous search capability and avoiding premature convergence.

IMF Inertia Weight Graph

3.2.3 Adaptive updating mechanism

The population updating mechanism based on IMF inertia weight effectively reduces the density of population clustering to a certain extent. However, due to the intrinsic dynamics of the GWO, the newly generated wolf packs are still inevitably concentrated and migrate towards the positions directed by the α -wolf, β -wolf, and δ -wolf during the iterative process. In response to this, the present study defines the aggregation coefficient as the ratio of an individual's fitness value to the average population fitness value, which serves to quantify the degree of divergence between the current solution and the optimal solution. In minimization problems, the smaller the fitness value, the better the solution. A smaller aggregation coefficient indicates a more favorable current solution, thus allowing for minor updates in the vicinity of the individual's current position. Conversely, a larger aggregation coefficient suggests a poor location of the individual, warranting a significant perturbation to facilitate a jump to other positions. Based on this analysis, this paper introduces a Sigmoid function to construct the adaptive updating amplitude of the population under different aggregation coefficients, as depicted in Eqs. ( 11 )-( 12 ).

where f i represents the fitness value of the i th individual, and f ave denotes the average fitness value of the population. θ is the exponential coefficient, which is taken as 0.5 in this paper.

In comparison to the standard GWO, the IAGWO brings several significant advancements. Firstly, it introduces a novel search mechanism by incorporating velocity concepts. This addition helps in preventing premature convergence and allows for a more thorough exploration of the search space. The integration of velocity updates also adds an element of randomness, which in turn increases the diversity within the population of solutions. Moreover, the implementation of the IMF inertia weight strategy in IAGWO improves the balance between exploring the global search space and exploiting local solutions. This strategic enhancement significantly boosts the convergence speed of the algorithm. Furthermore, IAGWO differentiates itself from the standard GWO through its adaptive updating mechanism. This mechanism combines the aggregation coefficient with the Sigmoid function, enhancing the algorithm's ability to switch between broad search patterns and detailed solution refinement. This results in improved performance in maintaining diversity and achieving faster convergence rates. This adaptive approach enables IAGWO to search and optimize more efficiently within the solution space of the problem. For an in-depth comprehension of the workings of IAGWO, the procedural flow is visually depicted in Fig.  4 , its pseudocode is meticulously detailed in Algorithm 2, and the proposed IAGWO workflow(Chauhan & Yadav 2023b ) is shown in Fig.  5 .

Implementation Process for IAGWO

Working procedure of the proposed IAGWO algorithm

3.3 Time complexity analysis

CEC17 (Competition on Evolutionary Computation) defines algorithm complexity as a measure of the computational resources required by an algorithm to solve a given problem instance. This section explains the computational complexity of IAGWO. The complexity of IAGWO is primarily influenced by two main factors: the initialization of solutions and the execution of the algorithm's core functions. These core functions involve calculating fitness functions and updating solutions. The computational complexity is determined by considering several variables: the count of solutions \(\left(N\right)\) , the upper limit of iterations \((T)\) , and the problem's dimension \((D)\) being tackled. Specifically, the complexity of initializing solutions in the IAGWO algorithm can be represented as \(O(N)\) , indicating the order of complexity in relation to the number of solutions. This gives an understanding of the computational resources required for the initial setup phase of the algorithm. This means that as the number of solutions \(N\) increases, the computational complexity of the initial solution of the algorithm will also increase accordingly. This represents the order of complexity for the initial setup phase of the algorithm. The time complexity of the original GWO algorithm is \(O(T\times N\times D)\) . IAGWO modifies this with Eqs. ( 8 ), ( 9 ), and ( 10 )–( 11 ), including enhancements to population diversity using the PSO position updating strategy, integration of IMF weights to reduce the excessive influence of higher-level wolves on lower-level ones, and the introduction of a population adaptive update based on the sigmoid function. The PSO position updating strategy requires calculations for each individual and each dimension, with a complexity of \(O(T\times N\times D)\) . The update from Eq. ( 10 ) is independent of population size and search dimensions, correlating only with the maximum number of iterations, resulting in a time complexity of \(O(T)\) . The time complexity for Eq. ( 11 ) is \(O(N\times D)\) . Consequently, the overall time complexity of IAGWO is \(O(\text{IAGWO})=O\left(T\times N\times D\right)+O\left(T\right)+O\left(N\times D\right)=O(T\times N\times D)\) , consistent with the original algorithm.

4 Results and comprehensive analysis

The simulation for this study was carried out on a Windows 11 platform, operating on a 64-bit system. The analysis was performed using MATLAB 2023b, running on a machine equipped with an AMD Ryzen 7 4800H CPU at 2.30 GHz and 16 GB of RAM.

4.1 Test functions and parameter settings

In this paper, the CEC 2017 (Dim = 30) (Mallipeddi and Suganthan 2010 ), CEC 2020 (Dim = 10 and 20) (Liang et al. 2019 ), and CEC 2022 (Dim = 10 and 20) (Ahrari et al. 2022) test suites were employed to evaluate the performance of the proposed IAGWO algorithm. The test suite for evaluating algorithms covers four different functional types: single-modal, multimodal, mixed, and combined. These different types of test suites are designed to comprehensively evaluate the performance and applicability of algorithms. Additionally, for assessing the scalability of the IAGWO algorithm, we employed the CEC 2013 Large-scale Global Optimization suite (800-dimensional) for simulation analysis (Li et al. 2013 ). The suite contains 15 highly complex reference functions that are grouped into four groups: fully separable, partially additively separable, overlapping, and completely indivisible. These different types of benchmark functions provide a comprehensive experimental framework for evaluating the scalability of optimization algorithms, so that we can more accurately evaluate the performance of IAGWO algorithm on different types of problems.

4.2 Comparison with other algorithms and parameter settings

The performance of the Improved Adaptive Grey Wolf Optimization (IAGWO) is benchmarked against 12 well-known algorithms, grouped into three categories for comparison:

High-citation algorithms: These include the Gravitational Search Algorithm (GSA) (Rashedi et al. 2009 ), Dolphin Echolocation Optimization (DMO) (Kaveh and Farhoudi 2013 ), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016 ), and Harris Hawks Optimization (HHO) (Tripathy et al. 2022 ).

Advanced algorithms: This category includes Combined Particle Swarm Optimization and Gravitational Search Algorithm (CPSOGSA), Crow Optimization Algorithm (COA) (Jia et al. 2023 ), African Vulture Optimization Algorithm (AVOA) (Abdollahzadeh et al. 2021 ), Optical Microscope Algorithm (OMA) (Cheng and Sholeh 2023 ), and Adaptive Artificial Electric Field Algorithm (iAEFA) (Chauhan and Yadav 2023a ).

GWO and its variants: This includes the original Grey Wolf Optimization (GWO), the Adaptive GWO (AGWO) (Meidani et al. 2022 ), the Enhanced GWO (ENGWO)(Mohammed et al. 2024 )and the Revised GWO (RGWO) (Banaie-Dezfouli et al. 2021 ).

Table 1 offers a comprehensive summary of the parameters for 14 different MH algorithms. For each of these algorithms, 30 independent runs were conducted, with each run limited to a maximum of 500 iterations and population size is set to 30 and a maximum of 30,000 evaluations. The outcomes of these runs were meticulously recorded, capturing the average values (denoted as Ave) and the standard deviations (Std) for each algorithm. To facilitate an easy comparison of their performance, the table highlights the best results among these 14 algorithms by formatting them in bold text. This highlighting method provides a clear visual indicator of which algorithms performed most effectively under the given testing conditions.

4.3 Qualitative assessment of IAGWO

4.3.1 exploring convergence patterns.

To verify the convergence performance of IAGWO, we plotted its convergence performance evaluation on the 30-dimensional CEC2017 test functions, as shown in Fig.  6 . It presents the corresponding results of different test functions in the form of nine images involving three instances selected from the same suite. In the presentation of images, the first column distinctly illustrates the two-dimensional profiles of the reference functions being analyzed. The visuals presented in the first column accurately depict the characteristics and contours of each function being optimized. These graphical representations offer a clear understanding of the challenges and intricacies of each function. In the second column, images depict the final positions of the search agents at the end of the optimization process. Within these visuals, the optimal solution's location is distinctly marked with a red dot. This not only illustrates the end point of the search agents' journey but also visually highlights the spot where they successfully identified the most favorable solution. This layout effectively communicates the results of the optimization process, making it easier to comprehend the behavior and efficacy of the search agents in navigating the solution space. This layout offers a clear and informative view of both the nature of the functions and the outcomes of the optimization process. By observing the second column of images, we can clearly find that the search agent is close to the optimal solution in most cases, which fully reflects the powerful ability of IAGWO algorithm in the process of exploration and development. In addition, the third column image accurately tracks the change of the average fitness value during the iteration. Initially high, these values decrease and stabilize after 100 iterations, albeit with minor fluctuations. These fluctuations are normal in complex optimization problems, indicate ongoing detailed searches for improvement and the maintenance of population diversity to prevent premature convergence to local optima. The fourth column reveals the search agents' trajectories in the first dimension, there were marked fluctuations in the early iterations, which then leveled off, and then fluctuations again at intervals, which leveled off again, signifying a balance between exploration and exploitation. Finally, the convergence curve, smooth for unimodal functions, suggests optimal values are achievable through iteration. For multimodal functions, however, the step-like curve reflects the need for continual avoidance of local optima to reach global optima. These four metrics collectively affirm IAGWO's robust convergence.

The convergence behavior of IAGWO

4.3.2 Analyzing the diversity of population

In optimization algorithms, the importance of population diversity is a matter of balance. Moderate population diversity can help the algorithm avoid falling into local optima, thereby increasing search space coverage and global search capability, improving convergence speed, and the quality of optimization results. However, excessive population diversity may lead to overly dispersed search, making it difficult for the algorithm to explore local regions deeply, thereby reducing convergence speed and the quality of final solutions. Therefore, when designing optimization algorithms, it is necessary to consider a balance between population diversity and search efficiency. This can be achieved through appropriate parameter settings or suitable strategies to maintain population diversity, thus effectively solving optimization problems. A population with high diversity indicates significant differences among individuals, allowing for broader exploration in the search space and avoiding premature convergence to local optima. Hence, maintaining good population diversity is a crucial objective in metaheuristic algorithms. Typically, we use Eq. ( 13 ) and Eq. ( 14 ) to measure the population diversity of the algorithm. This calculation method was proposed by Morrison in 2004. Where, \({I}_{C}\) represents the moment of inertia, \({x}_{id}\) denotes the ith search agent's value in the \({d}^{th}\) dimension at iteration t . Furthermore, \({c}_{d}\) represents the spread of the population from its center of mass, denoted by ' c ', in every iteration, as illustrated in Eq. ( 14 ) (Fu et al. 2023a , b).

Figure  7 displays the comparative experimental outcomes regarding population diversity for both IAGWO and GWO. The measurement of population diversity is conducted through \({I}_{C}\) . Observations from Fig.  7 reveal that IAGWO demonstrates an initial marked increase in diversity during the early phases of iteration, which then transitions to a state of relative stability at an elevated level. This indicates an increase in the variance among individuals within the IAGWO population during the early iterations, effectively exploring a vast search space. As iterations progress, the population diversity tends to stabilize, which aids in averting premature convergence to local optima. The minor fluctuation are normal and beneficial for the algorithm to adapt to dynamically changing search spaces and prevent premature convergence. In contrast, GWO shows insufficient population diversity, highlighting IAGWO’s effectiveness in maintaining diversity, crucial for exploring complex search spaces and avoiding local optima. These experimental outcomes demonstrate IAGWO's substantial potential in optimization.

The population diversity of IAGWO and GWO

4.3.3 Exploration and exploitation analysis

In optimization algorithms, managing the balance between exploration and exploitation is key for optimal performance (Saka et al. 2016 ). Exploration involves searching through the solution space, while exploitation focuses on refining known good solutions. This section deals with quantifying the extent of exploration and exploitation in the algorithm. To do this, we use Eq. ( 15 ) to calculate the percentage of exploration and Eq. ( 16 ) for the percentage of exploitation. Additionally, the parameter \(Div\left(t\right)\) used for measuring dimension diversity is calculated using Eq. ( 17 ). The parameter \({\rm Div}_{max}\) reflects the peak diversity noted throughout the entire course of iterations,, which is essential for understanding how broadly and effectively the algorithm explores the solution space(Li et al. 2023 ) (Nadimi-Shahraki et al. 2023 ).

Figure  8 depicts the results of the experiments conducted. It shows that for various function types, as the number of iterations progresses, GWO consistently demonstrates a higher rate of exploration and a comparatively lower rate of exploitation. In contrast, IAGWO shows a changing pattern, with exploration decreasing and exploitation increasing as iterations progress. This observation suggests that GWO tends towards a broad search across the entire space, with less focus on local search and weaker performance in thoroughly exploiting the optimal regions found. In comparison, IAGWO demonstrates the ability to dynamically adjust its search strategy. This implies that the algorithm initially identifies potential good solution areas through extensive exploration and then finely tunes these solutions in the later stages through focused exploitation, potentially enhancing both the efficiency of the algorithm and the quality of solutions. Overall, while GWO shows commendable exploration capabilities, it lacks effective exploitation. In contrast, IAGWO effectively strikes a balance between exploration and exploitation. This balance is well-maintained across a variety of benchmark functions, showcasing IAGWO's adaptability and efficiency in different optimization scenarios. This attribute is particularly important as it ensures the algorithm can thoroughly search the solution space while also honing in on the most promising solutions.

The exploration and exploitation of IAGWO and GWO

4.3.4 Ablation experiments

In this section, a detailed analysis is conducted on the impact of three proposed improvement strategies on the GWO. These strategies include the PSO position updating mechanism, the introduction of IMF inertia weight strategy, and the adoption of a Sigmoid adaptive updating strategy. Based on these improvements, three new algorithm variants are named: PGWO for the PSO search mechanism, IGWO for the IMF inertia weight, and SGWO for the Sigmoid adaptive updating strategy. According to the experimental results in Fig.  9 , all three strategies significantly enhance the convergence accuracy and speed of GWO, with IAGWO showing particularly notable performance.

Comparison of different improvement strategies

Specifically, when dealing with unimodal and multimodal functions, the results of PGWO and IAGWO are relatively consistent, showing a more significant improvement over GWO compared to SGWO and IGWO. However, when dealing with more complex hybrid modal functions, the enhancement of PGWO on GWO diminishes, while the IAGWO algorithm, integrating all three strategies, continues to exhibit exceptional optimization performance. Overall, the IAGWO algorithm successfully overcomes challenges of local optima and premature convergence, significantly boosting the algorithm's convergence speed and accuracy. These findings provide valuable insights for the further development and application of the GWO.

4.4 Quantitative evaluation

In this section, the efficacy of IAGWO is scrutinized using a series of test suites: CEC 2017, CEC 2020, and CEC 2022. Moreover, its proficiency in handling large-scale problems is assessed with the CEC 2013 suite. To clearly compare performance, the best results among the algorithms are highlighted in bold in the tables. The parameters are standardized with a population size of 100, a maximum iteration limit of 500, and a total of 30 independent runs. The performance outcomes are systematically presented in Tables  2 to 7 , which illustrate the average values (Ave) and the standard deviations (Std) for each competing algorithm. A thorough statistical analysis is conducted to highlight the superiority of IAGWO. This includes an initial evaluation represented by three indicators (W|T|L) in the first line of the results, denoting the algorithms' performance as best (win), comparable (tie), or least effective (loss) for specific functions. The second row compiles the mean performance of all algorithms, while the third row offers insights into the overall standings through the final Friedman ranking. The tables distinctly highlight the top results, emphasizing their significance. Furthermore, the comparative analysis of the convergence curves for each algorithm is depicted in Fig.  10 . This visual representation aids in understanding the progression and efficiency of each algorithm in finding optimal solutions over the course of iterations. This detailed evaluation underscores the robustness and adaptability of IAGWO in varied optimization contexts.

Convergence curves of different algorithms

4.4.1 Assessing performance with CEC 2017 test suite

This section examines the efficacy of IAGWO using the CEC 2017 test suite with a dimensionality of 30, as detailed in Table  2 . The results are quite telling: IAGWO recorded the highest number of best performances, leading in 16 out of the 30 functions tested. Notably, it did not register as the least effective in any of the functions. In terms of statistical standing, IAGWO's Friedman mean ranking is 3.00, earning it the top position. Further, a diverse range of functions from CEC 2017 (Dim = 30) were chosen for a more comprehensive evaluation. The comparative analysis of the convergence trends, depicted in Fig.  10 , reveals that IAGWO consistently achieved the quickest convergence rate and maintained the highest level of accuracy in convergence. These results underscore IAGWO's exceptional proficiency in both global exploration and local exploitation. Collectively, these findings solidify the effectiveness and superiority of IAGWO as an optimization tool.

4.4.2 Assessing performance with CEC 2020 test suite

This section is dedicated to evaluating 13 algorithms with the utilization of the CEC 2020 test suite, which includes tests with dimensions of 10 and 20. The outcomes of this evaluation are systematically presented in Table  3 and Table  4 . On the CEC 2020 tests, IAGWO mirrors the impressive results observed in the CEC 2017 suite, achieving the highest number of best performances while not being the least effective in any function. To provide a visual representation of these results, representative functions are chosen to illustrate the convergence curves, as depicted in Fig.  10 . IAGWO consistently shows the quickest convergence speed and the highest accuracy in convergence, reaffirming its efficiency. Additionally, it's important to note the contrasting performance of GWO on the CEC 2020 suite. Despite its lower ranking in the Friedman rankings, indicating a comparatively poor performance, its improved variants, namely AGWO, ENGWO, and RGWO, show marked improvements. Remarkably, RGWO secures the second-highest ranking, closely following IAGWO, underscoring the substantial research value in enhancing the GWO algorithm. A comprehensive statistical analysis among the 13 algorithms tested places IAGWO at the forefront in the Friedman rankings. This achievement highlights its superiority not only over the original GWO but also over other well-regarded algorithms. These results collectively demonstrate the robustness and effectiveness of IAGWO in a competitive algorithmic landscape.

4.4.3 Assessing performance with CEC 2022 test suite

This section is dedicated to a thorough evaluation of the proposed IAGWO and 12 other comparative algorithms, utilizing the CEC 2022 test suite. The primary objective of this evaluation is to gauge the exploration and exploitation capabilities of these algorithms and assess their proficiency in avoiding local optima traps. The experiments are conducted under 10-dimensional and 20-dimensional scenarios, with corresponding results displayed in Tables  5 and 6 , respectively. IAGWO ranks first in Friedman mean ranking in both dimensional settings, with ranking values of 1.75 and 2.25 respectively. Similarly, while GWO shows subpar performance, its variants enhance GWO's performance, emphasizing the research significance of GWO. The analysis of results depicted in Fig.  10 leads to a conclusive observation that IAGWO successfully evades getting stuck in local optima and avoids premature convergence. These findings serve not just as a testament to the excellence and robustness of IAGWO, but they also highlight its substantial performance benefits and the capability to yield enhanced solutions. This analysis underscores IAGWO's effectiveness in navigating complex optimization landscapes, further establishing its potential as a superior tool in optimization tasks.

4.4.4 Scalability evaluation using the CEC 2013 test suite

In real-world scenarios, solving optimization problems often requires adjusting multiple parameters at once. To test the scalability of the IAGWO for high-dimensional problems, we utilized the CEC 2013 suite for large-scale global optimization. The results of this testing are detailed in Table  7 . This suite includes 15 highly complex test functions, each with up to 1000 dimensions, providing a robust challenge for assessing algorithmic performance. In our experiments, IAGWO was compared with 12 other algorithms. The population size was fixed at 100, and we limited the maximum number of iterations to 10 for each run. After conducting 30 independent runs for each algorithm, IAGWO achieved a Friedman mean rank value of 2.63. This score signifies a higher level of performance relative to the other algorithms in the competition. The findings from these experiments demonstrate that the IAGWO algorithm has significant scalability, effectively handling complex, high-dimensional optimization challenges. This capability distinguishes IAGWO from other algorithms, highlighting its suitability for practical, large-scale optimization applications.

4.5 Wilcoxon rank sum test

This study utilizes the non-parametric Wilcoxon rank sum test (Wilcoxon 1945 ) to conduct comparative performance assessments of various algorithms, setting the significance level at 0.05. To succinctly represent the performance of IAGWO relative to its competitors, the symbols “ + / = /-” are used to denote whether IAGWO is superior to, equivalent to, or inferior to the competing algorithms. As shown in Table  8 , these statistical results clearly indicate significant performance differences between IAGWO and other competing algorithms in most cases. Specifically, the statistical data show the following comparative results: 344/0/46、119/0/11、111/13/6、150/0/6、152/0/4 and 175/0/5. The analysis presented above demonstrates that the IAGWO method, as introduced in this study, shows exceptional overall performance when compared to the traditional GWO and other rival algorithms, thereby underscoring its distinct advantages.

4.6 Time comparison analysis of IAGWO and GWO

Building on the findings from previous chapters, it's clear that IAGWO significantly surpasses the original GWO in terms of overall performance. In this section, we focus on a more detailed comparison, specifically looking at the computational costs of both algorithms, with a particular emphasis on the differences in computational time. To facilitate this comparison, we standardized the settings for both IAGWO and GWO. For this evaluation, the population size was configured to 50, the maximum iterations were limited to 1000, and each algorithm underwent 30 independent runs. Table 9 presents the total time (in seconds) each algorithm took to complete all 30 runs. This data provides a clear basis for comparing the efficiency of the two algorithms in terms of how long they take to execute, offering insights into their time-based performance efficiency.

Analysis of the experimental data on the CEC 2017 test suit (Dim = 30) indicates that under the same experimental parameters, IAGWO and GWO perform almost equally in terms of execution time on unimodal functions and some simpler multimodal functions, but when dealing with more complex multimodal and hybrid functions, IAGWO generally consumes significantly less computational time than GWO. This suggests that in handling highly complex problems, IAGWO demonstrates greater computational efficiency. Compared to the original GWO, IAGWO's improved search strategies are more efficient, possessing better global search capabilities or faster local convergence speeds. Overall, IAGWO not only excels in benchmark tests but also exhibits higher computational efficiency and better adaptability when addressing more complex optimization problems that may arise in practical applications.

However, on the CEC 2020 test suite (Dim = 10 and 20) and CEC 2022 test suite (Dim = 10 and 20), IAGWO generally exhibits higher computational times compared to GWO. This may indicate that the types of problems or characteristics included in CEC 2020 are not entirely compatible with the strategies of IAGWO, leading to a higher computational load.

4.7 Evaluating performance against CEC 2014 and CEC 2017 competition-winners

This section evaluates the performance of the proposed IAGWO using the CEC 2014 test suite with a dimensionality of 30 (Liang et al. 2013 ) and CEC 2017 (Dim = 30) test suites. Additionally, we compare the performance of IAGWO with the competition-winners of these two suites in previous CEC competitions, including L-SHADE (Tanabe & Fukunaga 2014 ) and AL-SHADE (Li et al. 2022 ) from CEC 2014, and LSHADE-SPACMA (Mohamed, Hadi, Fattouh, & Jambi, 2017) and LSHADE-cnEpSin (Awad, Ali, & Suganthan, 2017) from CEC 2017. In the experimental setup, the population size is fixed at 30, the maximum iterations are limited to 500, and a total of 30 independent runs are performed.

Table 10 presents the results from testing IAGWO using the CEC 2014 suite. In these tests, IAGWO surpassed other algorithms in six different scenarios, though it showed slightly weaker performance in one. Notably, IAGWO achieved a Friedman mean ranking value of 1.71, which places it second after L-SHADE but ahead of AL-SHADE. Table 11 focuses on the performance of IAGWO in the CEC 2017 suite. Here, IAGWO showed strong results in 8 of the test cases, but its performance was less impressive in 10 others. In terms of the Friedman mean ranking, IAGWO scored 1.99, which is slightly better than LSHADE-SPACMA, but not quite as good as LSHADE-cnEpSin. These results provide a detailed comparison of IAGWO's performance relative to other algorithms in these specific test environments.

Combining experimental outcomes, IAGWO can be positioned as a high-performing optimizer in test functions. These results not only demonstrate IAGWO's strong capability in handling different types of optimization problems but also indicate its competitive standing against existing top-tier algorithms. These findings emphasize the potential application value of IAGWO in the field of evolutionary computing and optimization. This simultaneously demonstrates the effectiveness of the three improvement strategies we introduced: the PSO Search Mechanism, the IMF Inertia Weighting Strategy, and the Adaptive Updating Mechanism, enhancing the optimization performance of the algorithm.

4.8 IAGWO for 19 engineering design challenges

The specific constrained handling technique used in engineering design challenges is called "constraint relaxation." Constraint relaxation involves temporarily easing or loosening certain constraints within the design problem to explore alternative solutions. This allows designers to generate a wider range of potential solutions without being overly restricted by strict constraints. Once various solutions have been identified, designers can then reintroduce and refine the constraints to ensure that the final design meets all necessary requirements. Intelligent optimization algorithms can efficiently explore the design space and uncover potential solutions. By integrating constraint relaxation techniques, these algorithms can dynamically handle constraints during the search process, allowing for a broader exploration of the design space and enhancing the efficiency of finding optimal solutions.

In this section, the proficiency of IAGWO is meticulously evaluated through a set of 19 engineering design challenges (EDC) sourced from the CEC 2020 real-world optimization benchmarks, as outlined by Kumar et al., 2020., 2020. A concise summary of these engineering challenges is presented in Table  12 , which includes key details such as their dimensions ( D ), the count of inequality constraints ( g ), equality constraints ( h ), and the known optimal cost ( f min ). The evaluation parameters are defined as follows: a population size of 50, a maximum of 1000 iterations, and 30 independent runs for each challenge.

Table 13 is dedicated to enumerating the performance metrics of IAGWO. This table encompasses various metrics including the best cost achieved (Best), the average cost (Ave), the cost's standard deviation (Std), and performance symbols (W|T|L), representing the number of wins, ties, and losses, respectively. Additionally, the evaluation includes a comprehensive analysis of the mean performance of all the algorithms involved in the testing. It also presents a ranking of these methods, providing a clear and structured comparison of their overall effectiveness and highlights instances where IAGWO achieves optimal results.

The statistical analysis drawn from these results clearly demonstrates IAGWO's superior ability in solving these real-world engineering design challenges, effectively outshining other methods. In terms of overall effectiveness, other algorithms like OMA, DMO, RGWO, and ENGWO trail behind IAGWO. This comprehensive analysis accentuates the robustness and efficacy of IAGWO.

5 Summary and future directions

In this study, we introduced an enhanced version of the GWO, aiming to tackle its inherent limitations and elevate its efficacy for addressing contemporary optimization challenges. The original GWO, while promising, exhibited deficiencies, notably in its convergence speed and its adaptability to intricate, high-dimensional problem landscapes. To fortify its capabilities, we embarked on an innovative path, culminating in the birth of an enhanced variant dubbed the Improved Adaptive Grey Wolf Optimizer (IAGWO). Central to our enhancement strategy was the infusion of concepts borrowed from Particle Swarm Optimization (PSO), introducing a velocity component to expedite convergence. This integration of velocity mechanics injected dynamism into the algorithm, enabling it to traverse solution spaces with greater agility. Moreover, a novel search mechanism was devised, augmenting the algorithm's exploration and exploitation capabilities to navigate complex problem domains more efficiently. In addition to these fundamental alterations, we devised novel strategies for Inertia Weighting and Position Updating, leveraging Nonlinear Inertia Weighting for Intermediary Fitness (IMF) and employing Sigmoid adaptive techniques. These refinements were meticulously crafted to synergize with the core algorithm, amplifying its prowess in navigating diverse optimization landscapes with finesse.

To validate the prowess of IAGWO, rigorous experimentation ensued, wherein 52 test functions sourced from prestigious benchmark suites were scrutinized. Comparative analysis against eight prominent Metaheuristic (MH) algorithms, including the original GWO and three of its variants, underscored IAGWO's supremacy in terms of convergence speed and solution precision. Furthermore, the algorithm's mettle was tested against 15 formidable large-scale global optimization challenges, affirming its adeptness in grappling with high-dimensional complexities. The litmus test for IAGWO's efficacy extended to competitive arenas, where it stood toe-to-toe against previous champions of the renowned CEC competitions across various iterations. Notably, IAGWO's performance surpassed expectations, firmly establishing its dominance and resilience in the face of formidable adversaries. Beyond the realm of academia, the real-world applicability of IAGWO was validated through its deployment in 19 diverse engineering design challenges. Here, its versatility and competitive edge shone brightly, outperforming established algorithms and offering tangible solutions to practical problems.

Despite its commendable achievements, the journey of IAGWO is far from over. While it has emerged as a potent force in the optimization landscape, ongoing efforts are directed towards fine-tuning its computational efficiency. Time comparison analyses have revealed areas for optimization, particularly concerning computational overhead in certain test suites. Future endeavors will thus focus on streamlining computational complexity without compromising search efficacy, ensuring that IAGWO remains at the forefront of optimization methodologies. Looking ahead, the horizon for IAGWO is brimming with promise. Beyond academic benchmarks, its utility extends to a myriad of real-world applications, ranging from feature extraction to operations research, classification, and logistical challenges. As we embark on this journey, our aim is clear: to harness the full potential of IAGWO in unraveling the complexities of the modern world, one optimization problem at a time.

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This work was supported by Natural Science Foundation of Tianjin Municipality (21JCYBJC00110) and China Postdoctoral Science Foundation (2023M731803).

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Mingyang Yu, Jing Xu, Weiyun Liang, Yu Qiu & Sixu Bao

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MY: conceptualization, methodology, writing—original draft, formal analysis, data curation, writing—review & editing, software. WL: visualization, formal analysis, writing—review & editing. JX: conceptualization, resources, supervision, formal analysis. YQ: software, writing—review & editing, resources. SB: methodology, visualization resources, software. LT: visualization resources.

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Yu, M., Xu, J., Liang, W. et al. Improved multi-strategy adaptive Grey Wolf Optimization for practical engineering applications and high-dimensional problem solving. Artif Intell Rev 57 , 277 (2024). https://doi.org/10.1007/s10462-024-10821-3

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Accepted : 28 May 2024

Published : 05 September 2024

DOI : https://doi.org/10.1007/s10462-024-10821-3

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