year 4 problem solving addition

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Year 4 Maths Worksheets UK Hub Page

Welcome to our Year 4 Maths Worksheets Hub page.

Here you will find our selection of printable maths worksheets for Year 4 children, for your child will enjoy.

Take a look at our times table colouring pages, or maybe some of our fraction of shapes worksheets. Perhaps you would prefer our time worksheets, or learning about line or block symmetry?

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• This page contains links to other Math webpages where you will find a range of activities and resources.
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Year 4 Maths Learning

Here are some of the key learning objectives for the end of Year 4:

• know and use Place value up to 4 digits
• Compare and order numbers up to 10,000
• Counting on and back in 1s, 10s, 100s and 1,000s from different starting points
• Position numbers on a number line up to 10,000
• Round numbers to the nearest 10, 100 or 1000.
• Count backwards through zero and use negative numbers.
• add and subtract with up to 4 digits in columns
• add or subtract 1s, 10s, 100s or 1000s from a 4-digit number
• solve 2-step problems using addition and subtraction
• recall and use multiplication and division facts up to 12x12
• recognise and use factor pairs
• multiply 2-digit and 3-digit numbers by a 1-digit number
• solve problems using multiplication and division
• count up and down in hundredths
• recognise and use equivalent fractions
• add and subtract fractions with the same denominator
• solve fraction problems including with non-unit fractions
• write common fractions such halves and quarters as decimals;
• understand tenths and hundredths as decimals
• round decimals with 1dp to the nearest whole
• compare numbers with up to 2dp
• solve simple money and measure problems
• measure, compare and calculate using different measures
• find the area and perimeter of squares and rectangles
• convert between 12- and 24-hour clock
• convert between different units of measure
• identify and order acute and obtuse angles
• compare and classify 2D and 3D shapes according to their properties
• identify lines of symmetry in 2D shapes
• use coordinates in the first quadrant
• translate shapes up, down, left and right
• interpret and present data in bar graphs, pictograms and tabels
• solve 1-step and 2-step problems using data in tables, pictograms and graphs

Please note:

Our site is mainly based around the US Elementary school math standards.

Though the links on this page are all designed primarily for students in the US, but they are also at the correct level and standard for UK students.

The main issue is that some of the spelling is different and this site uses US spelling.

Year 4 is generally equivalent to 3rd Grade in the US.

On this page you will find link to our range of math worksheets for Year 4 pupils.

Quicklinks to Year 4 ...

• Online 3rd Grade Practice
• Place Value Zone
• Mental Math Zone

Word Problems Zone

Fractions zone.

• Measurement Zone

Geometry Zone

Data analysis zone.

• Fun Zone: games and puzzles

Coronavirus Stay At Home Support

For those parents who have found themselves unexpectedly at home with the kids and need some emergency activities for them to do, we have started to develop some Maths Grab Packs for kids in the UK.

Each pack consists of at least 10 mixed math worksheets on a variety of topics to help you keep you child occupied and learning.

The idea behind them is that they can be used out-of-the-box for some quick maths activities for your child.

They are completely FREE - take a look!

• Free Maths Grabs Packs

Place Value & Number Sense Zone

Year 4 numbers & place value worksheets.

Using these Year 4 maths worksheets will help your child to:

• learn their place value with 4 digit numbers;
• use place value models to understand how to combine thousands, hundreds, tens and ones;
• understand the value of each digit in a 4 digit number;
• learn to use standard and expanded form with 4 digit numbers.
• learn to read and write Roman numerals
• Place Value Models 4 Digits
• Place Value 4 Digit Numbers Worksheets (conversion)
• Ordering 4-Digit Numbers
• Ordering Negative Numbers -10 to 10
• Roman Numerals worksheets

Year 4 Counting & Sequences Worksheets

Each worksheets consists of a sequence which has been partially filled in. The rest of each sequence must be completed.

At this grade, the focus is on counting on and back in constant steps of a digit.

• Counting on and back by digits

Rounding, Inequalities, Multiples and Balancing Equations

Using these Year 4 Maths worksheets will help your child to:

• round a number to the nearest 10, 100 or 1000;
• use the > and < symbols correctly for inequalities;
• use multiples and apply them to solve problems.
• learn to balance math equations
• Rounding to the nearest 10 Worksheets
• Rounding to the nearest 100 worksheets
• Rounding to the nearest 1000 worksheets
• Rounding Inequalities Multiples Worksheets
• Balancing Math Equations

Year 4 Mental Math Zone

Here you will find a range of printable Year 4 mental maths quizzes for your child to enjoy.

Each quiz tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math quiz!

• Year 4 Mental Maths Test sheets

Number Bonds Worksheets

The worksheets on this page will help to develop children's knowledge of numbers bonds to 20, 50 and 100.

There are a range of matching and wordsearches for children to enjoy whilst developing their number bond knowledge.

• Number Bonds to 50 and 100

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Year 4 Addition Worksheets

• learn to add numbers mentally to 100;
• add on 1, 10, 100 and 1000 to different numbers;
• learn to add 4 digit numbers in columns;
• Addition Facts Worksheets to 100+100
• 4-Digit Addition Worksheets
• Money Addition Worksheets (£ )

Year 4 Subtraction Worksheets (3rd Grade)

Using these subtraction worksheets will help your child to:

• learn to subtract numbers mentally to 100;
• learn to do 4 Digit column subtraction.
• Third Grade Subtraction Worksheets to 100
• 4 Digit Subtraction Worksheets
• Money Subtraction Worksheets UK (£ )

Year 4 Multiplication Worksheets

• learn their multiplication tables up to 12 x 12;
• understand and use different models of multiplication;
• solve a range of Year 4 Multiplication problems.

Online Times Table Practice

• Times Tables Practice Zone

Understanding Multiplication

• Understanding Multiplication Facts Worksheets to 10x10

Multiplication Table Worksheets

• Multiplication Table Worksheets - 2 3 4 5 10
• Multiplication Drill Sheets 6 7 8 9
• Fun Multiplication Worksheets to 10x10
• Times Table Worksheets Circles 1 to 12 tables
• Multiplying (integers) by 10 and 100 Worksheets
• Multiplying by Multiples of 10 and 100

2-Digit Multiplication

• 2 Digit Multiplication Worksheets

Multiplication Word Problems

• Year 4 Multiplication Word Problem Worksheets (3rd Grade)

Randomly Generated Multiplication Worksheets

Using our random worksheet generator, you can:

• Choose the tables you want to test;
• Choose how big you want the numbers to go - up to 5 times, 10 times or bigger!
• Choose how many questions per page.
• Times Tables Worksheets (randomly generated)
• Free Multiplication Worksheets (randomly generated)
• Single Digit Multiplication Worksheets Generator
• Multiplication & Division Worksheets (randomly generated)

Year 4 Division Worksheets

Using these Year 4 Maths worksheets will help your child learn to:

• understand how division and multiplication relate to one another;
• know their division facts to 10x10;
• begin to learn 2-digit by 1-digit long division.
• Division Facts to 10x10 Worksheets
• Divding by Multiples of 10 and 100 Worksheets
• Year 4 Long Division Worksheets (3rd grade)
• Division Facts Worksheets (randomly generated)

Using the 3rd Grade Math worksheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• develop their knowledge of fractions;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

These sheets involve solving one or two more challenging longer problems.

• Year 4 Math Problems (3rd Grade)

These sheets involve solving many 'real-life' problems involving data.

• Year 4 Math Word Problems for kids (3rd Grade)

These sheets involve solving 3-digit and 4-digit addition word problems.

• Addition Word Problems 3rd Grade (3- and 4-digits)

These sheets involve solving 3-digit and 4-digit subtraction problems.

• Subtraction Word Problems 3rd Grade

These sheets involve solving a range of multiplciation problems.

These sheets involve solving a range of division problems.

• Division Worksheets Grade 3 Word Problems

Year 4 Fraction Worksheets

Using these sheets will help your child to:

• understand what fractions are;
• relate fractions to everyday objects and quantities;
• place different fractions on a number line;
• shade in different fractions of a shapes;
• work out unit fractions of numbers.

• What is a Mixed Number Support page
• Finding Fractions - Fraction Spotting
• Fractions of Shapes Worksheets
• Unit Fraction of Numbers
• Halves and Quarters (up to 100)
• Fraction Number Line Sheets
• Adding Fractions with Like Denominators
• Subtracting Fractions with like denominators
• Fraction Riddles for kids (easier)

Year 4 Geometry Worksheets

The following worksheets will help your child to:

• Identify and name a range of 2d and 3d shapes;
• Draw 2d shapes;
• Use reflective symmetry to reflect shapes in a mirror line.
• recognise and identify right angles and lines of symmetry;
• recognise and identify parallel lines;
• identify the faces, edges, vertices and nets of 3d shapes;
• Year 4 Free Printable Geometry Worksheets
• Block Symmetry Worksheet
• Line Symmetry Worksheets
• Symmetry Activities
• Geometry Nets Information and Worksheets

Measurement Zone, including Time & Money

Year 4 measurement worksheets.

• Year 4 Measurement Worksheets - reading scales
• Metric Conversion Worksheets

Year 4 Money Worksheets

Using challenges is a great way to get kids to use their thinking skills and extend learning by applying the knowledge they have.

• count a range of coins up to £10
• compare money amounts
• apply their existing skills to puzzle out clues;
• understand money terminology;
• develop their thinking skills.
• Year 4 Money Challenges
• Column Addition Money Worksheets (UK)
• Column Subtraction Money Worksheets (UK)

Area and Perimeter Worksheets

• understand area and perimeter;
• learn how to find the area and perimeter of rectangles.
• Area Worksheets
• Perimeter Worksheets

Time Worksheets

Using the sheets in this section will help your child to:

• tell the time to the nearest 5 minutes;
• become familiar with both digital and analogue times;
• use the words 'past' and 'to' to describe the time correctly.
• add and subtract time intervals from times and work out time intervals.
• Add and Subtract Time Worksheets
• Elapsed Time Worksheets
• Printable Time Worksheets - Time Puzzles (easier)
• 24 Hour Clock Conversion Worksheets

On this page there are a selection of bar and picture graphs, including bar graphs with real-life data such as tree heights.

• Year 4 Bar Graph Worksheets (3rd grade)
• Year 4 Venn Diagram Worksheets

Fun Zone: Puzzles, Games and Riddles

Year 4 maths games.

The following games involve different Year 4 Maths activities which you and your child can enjoy together.

• Year 4 / Third Grade Math Games

Year 4 Math Puzzles

The puzzles will help your child practice and apply their addition, subtraction and multiplication facts as well as developing their thinking and reasoning skills in a fun and engaging way.

• Year 4 Math Puzzle Worksheets (3rd Grade)

Math Salamanders Year 4 Maths Games Ebook

Our Year 4 Maths Games Ebook contains all of our fun maths games, complete with instructions and resources.

This ebooklet is available in our store - use the link below to find out more!

• Year 4 Maths Games Ebook

Other UK Maths Worksheet pages

See below for our other maths worksheets hub pages designed for children in the UK.

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Roll the dice, make four-digit numbers, and round them to the nearest 10.

4. Use lots of different methods

Encourage your child to explore a range of methods for solving addition and subtraction problems.

Methods could include partitioning numbers into parts to add or subtract. For example:

2143 + 625 = ?   To solve this, you could partition 2143 into 2000, 100, 40, and 3, and then partition 625 into 600, 20, and 5.   The next step is to add the numbers together. First, add the ones: 5 + 3 = 8. Then, add the tens: 40 + 20 = 60. Then, add the hundreds: 600 + 100 = 700. And you also have the 2000.   Then, you can add all of these together: 8 + 60 + 700 + 2000 = 2768.

Your child may also draw pictures to represent how they have added or subtracted numbers. Number lines and number grids are useful for solving problems, as are formal written methods like column addition and subtraction.

Your child will understand subtraction as ‘difference’ as well as ‘taking away’. A good method that sees subtraction as difference is placing groups of objects into two rows to compare them and find the difference. This is particularly good for more tactile learners.

Your child may also find the difference between two numbers by counting up or counting back. For example, 27 – 18 could be interpreted as, ‘What is the difference between 27 and 18?’ Your child may count back from 27 to 18 to find the difference of 9 or count up from 18 to 27 to find the difference of 9.

Practising lots of methods not only means your child can more easily check their work – it means they can always pick the best method for any particular question. When your child has solved a problem, encourage them to use a different strategy to check their answer, and ask why they have chosen that particular method.

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Year 4 – Addition and Subtraction

Welcome to Year 4 Addition and Subtraction Value at Primary Maths Hub. Here you will find a growing library of outstanding resources and activities to support addition and subtraction lessons in Year 4 and at home.  

If there’s a resource you’d like to see here, just visit our ‘Request a Resource’ page and Primary Maths Hub will create the resource and add it to the site.

Do Now Tasks / Starters

Addition to 1,000 Do Now Tasks

Subtraction to 1,000 Do Now Tasks

Addition and Subtraction to 1,000 Do Now Tasks

Differentiated question sets.

Addition Question Set

Subtraction Question Set

Column Addition

4-Digit Column Addition

Faded Scaffolding Strips- 4-Digit Addition

4-Digit Column Addition –  No Re-Grouping

Faded Scaffolding Strips- 4-Digit Addition – No Regrouping

Faded Scaffolding Strips –  Adding Numbers  to 10,000

Adding Numbers to 10,000

Column Addition With PV Counters- One Regrouping

Subtraction.

Column Subtraction

4-Digit Column Subtraction

Faded Scaffolding Strips- 4-Digit Subtraction

4-Digit Column Subtraction – No Re-Grouping

Faded Scaffolding Strips- 4-Digit Subtraction- No Regrouping

Faded Scaffolding Strips- Subtracting From Multiples of 1,000

Faded Scaffolding Strips –  Subtracting Numbers  to 10,000

Subtract From Multiples  of 100 to 10,000

Strips- Subtracting Numbers to 10,000

Column Subtaction With PV Counters- One Regrouping

Mixed addition and subtraction.

4-Digit Mixed Column –  Addition and Subtraction

Missing Number –  Addition and Subtraction

Mixed Column Addition and Subtraction

Faded Scaffolding Strips- 4-Digit Mixed Addition and Subtraction

Faded Scaffolding Strips- Mixed Addition and Subtraction to 10,000

Strips- Adding and Subtracting Numbers to 10,000

Word problems and challenges.

Addition Word Problems  to 10,000

Challenge 1

Addition and Subtraction Word Problems to 10,000

Subtraction Word Problems to 10,000

Equality and inequality statements.

Equality and Inequality Statement Matching

Resources to support addition and subtraction.

Layout Support

Steps to success.

Column Addition Up To 4 Digits – Steps to Success

Column Subtraction Up To 4 Digits – Steps to Success

Subtract From Multiples of 100 – Steps to Success

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Free Printable Addition Word Problems Worksheets for 4th Year

Addition Word Problems: Discover a collection of free printable worksheets for Year 4 students, focusing on math addition word problems. Enhance learning and problem-solving skills with Quizizz.

Temas recomendados para ti

• Two-Step Word Problems
• Two-Digit Addition Word Problems

Explorar Addition Word Problems hojas de trabajo por grados

• kindergarten

Explorar Addition Word Problems Hojas de trabajo para year 4 por Tema

Explore otras hojas de trabajo de materias para year 4.

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Explore printable Addition Word Problems worksheets for 4th Year

Addition Word Problems worksheets for Year 4 are an excellent resource for teachers looking to help their students improve their math skills. These worksheets provide a variety of math word problems that are specifically designed for fourth-grade students, ensuring that the content is both age-appropriate and engaging. By incorporating these worksheets into their lesson plans, teachers can effectively teach important math concepts while also helping their students develop critical thinking and problem-solving skills. With a wide range of problems and scenarios, these Year 4 math worksheets are perfect for reinforcing classroom learning and providing extra practice for students who may be struggling with addition word problems.

In addition to using Addition Word Problems worksheets for Year 4, teachers can also take advantage of Quizizz, an online platform that offers a variety of interactive quizzes and activities to enhance student learning. Quizizz allows teachers to create custom quizzes, which can be tailored to the specific needs of their students, including math word problems for fourth graders. This platform also offers a wide range of other educational resources, such as interactive games, flashcards, and more, making it an invaluable tool for teachers looking to diversify their lesson plans and engage their students in new and exciting ways. By incorporating both Year 4 math worksheets and Quizizz into their teaching strategies, educators can ensure that their students receive a well-rounded education and develop the skills necessary for success in math and beyond.

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Addition Word Problem Worksheets

The addition word problem worksheets presented here involve performing addition operations with regrouping and without regrouping. Our extensive and well-researched word problem worksheets feature real-life scenarios that involve single-digit addition, two-digit addition, three-digit addition, and addition of large numbers. These pdf handouts are designed to provide ample practice for elementary school children. Free worksheets are included.

Single-digit Addition Word Problems

These printable practice worksheets involve simple addition of single-digit numbers. Read the word problems and perform addition operations to arrive at the answers.

• Download the set

Addition Word Problems: Sum up to 20

Featured in these worksheets are engaging word problems whose sums add up to 20. Addends may have a combination of single-digit and two-digit numbers.

Addition Problems: Two-digit and Single-digit

A number of real-life scenarios in the form of word problems featured in the addition worksheets here involve single digit and two-digit addends.

Two-digit Addition Problems - No Regrouping

The word problems in this section do not require regrouping or carrying. Find the answers to the word problems that feature two-digit addends.

Two-digit Addition Problems - With Regrouping

All two-digit addition word problems presented in this set of worksheets here require regrouping (carry over). Follow the place value columns to sum up the two-digit addends.

Theme based Word Problems

Presented here are worksheets with three colorful themes - Fall Season, Aquarium and Theme Park. Read the questions and solve the word problems. Answer keys are included.

Three-digit and Two-digit Addition

A total of 15 addition word problems spread over three PDF worksheets presented here require you to sum up three-digit addends with the two-digit addends.

Three-digit Addition Word Problems

Enhance your arithmetic skills. Read the word problems and sum up three-digit addends in these printable worksheets. Some problems may require regrouping. Answer key included in each worksheet.

Multi-digit Addition Word Problems: Adding Large Numbers

The word problems presented in the worksheets here feature large numbers with addends up to eight digits.

Related Worksheets

» Subtraction Word Problems

» Multiplication Word Problems

» Division Word Problems

» Math Word Problems

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Addition and subtraction word problems

Subject: Mathematics

Age range: 7-11

Resource type: Lesson (complete)

Last updated

11 April 2017

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Useful word problems.

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20 Word Problems For Year 4: Develop Their Problem Solving Skills Across Single and Mixed KS2 Topics

Emma Johnson

Word problems for Year 4 play an important role in Year 4 maths. In Year 4, the main focus is to ensure that pupils are becoming more fluent with whole numbers and the four operations. Students work to develop efficient written methods and to be accurate with their calculations. Pupils in Year 4 are exposed to a wider range of problem-solving questions and progress from one to two-step problems.

It is important that all children are given regular opportunities to access reasoning and word problem style questions. Fluency, reasoning and problem solving should be intertwined through every lesson, with all children having the opportunity to tackle each of these question types. 

All Kinds of Word Problems Multiplication

Strengthen your students' problem solving and multiplication skills with this pack of multiplication word problems

Place value 

Addition and subtraction , multiplication and division, fractions, decimals and percentages, measurement, why are word problems important in year 4 maths, how to teach problem solving in year 4, addition word problems for year 4, subtraction word problems for year 4, multiplication word problems for year 4, division word problems for year 4, fraction and decimal word problems in year 4, time word problems in year 4, multi-step word problems in year 4., more primary word problems resources.

There can sometimes be a tendency for reasoning and problem solving questions to be treated as extension activities for only the higher attaining pupils to attempt, but children of all abilities need to be accessing them on a regular basis.

To help you with this, we have put together a collection of 20 word problems aimed at Year 4 pupils. For more Year 4 maths resources, take a look at our collection of Year 4 maths worksheets .

Year 4 Maths Word Problems in the National Curriculum

In Year 4, pupils progress from solving one-step problems, to also being exposed to two-step problems across a range of topics, as set out in the National Curriculum.

Solve word problems involving counting in multiples of 6,7,9, 25 and 100; finding 1000 more or less than a given number; counting backwards through 0 to include negative numbers; ordering and comparing numbers beyond 1000 and rounding numbers to the nearest 10. 100 and 1000

Solve addition and subtraction word problems with up to 4 digits, including two-step word problems, deciding which operations and methods to use and why.

Solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems.

Solve problems involving increasingly harder fractions word problems to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number. Also Solve simple measure and money problems involving fractions and decimals to 2 decimal places.

Solve problems involving converting from hours to minutes, minutes to seconds, years to months, weeks to days.

Solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs.

Word problems are increasingly important as pupils move through Key Stage 2. As they become more confident with some of the core concepts pupils need to be applying this knowledge to a range of situations. By the end of Year 4, pupils should have memorised their multiplication tables up to and including the 12 times table and should be showing precision and fluency in their work.

Word problems in Year 4 should be fun and engaging for students. There are many ways to do this, including:

• acting out the problem;
• using manipulatives and visual images to help children understand the maths within the problem;
• use of talk partners to encourage children to discuss the question and share strategies for reaching a solution;
• using relatable problem solving situations.

Children need to be encouraged to read word problem questions carefully, to ensure they have identified the key information needed to be able to solve the problem. Pupils need to think about what they already know and how that information can help them to answer the question. They should also be encouraged to draw pictures and visual images, where appropriate, to help them to understand what the question is asking.

Here is an example:

A shop has an 8m roll of fabric.

The first customer buys 125cm of fabric and the second customer buys 3m from the same roll.

How much fabric is left on the roll, once the two customers have taken theirs?

How to solve:

What do you already know?

• The amount the first customer buys is given in cm, the amount the second customer buys is given in m. These needed to be converted to the same unit.
• Pupils in Year 4 need to be able to convert cm to m and vice versa. In this question, both the cm can be given as m or the m changed to cm to solve it.
• Once the units are the same, the two amounts need to be added together, to work out the total amount bought by the two customers.
• We can see this is a two-step question. To calculate how much fabric is left on the roll, the total amount bought by the two customers needs to be subtracted from the initial amount of fabric on the roll.

How can this be drawn/represented pictorially?

We can draw a bar model to represent this problem:

• To calculate the total amount of fabric bought, we need the units to be the same. We can either calculate in cm (300cm + 125cm = 425cm) or in m (3m + 1.25m = 4.25m)
• The amount of material bought needs to be subtracted from the original amount. This can again be solved in m or cm. Either in m: 8m – 4.25m = 3.75m or in cm: 800cm – 425cm = 375cm
• The total amount of material left is 3.75m or 375cm

In Year 4, addition word problems involve questions up to 4-digit numbers. They can include one and two-step addition and incorporate a range of concepts, such as measures and money word problems  

Addition question 1

It is 4164 miles to travel from London to Doha and 3266 miles to travel from Doha to Bangkok.

How far is it to travel from London to Bangkok, if the flight stops in Doha first?

Answer (1 mark): 7430 miles

Addition question 2

Fill in the missing numbers in this calculation.

Answer (1 mark): 6840

Addition question 3

On Saturday, 5486 fans attended a football game and 3748 fans attended a rugby game.

How many fans watched the two games in total?

Answer (1 mark): 9234

Subtraction word problems in Year 4, also involve numbers up to 4-digits, including both one and two-step problems, covering a range of concepts. By this stage, children should be confident in estimating and using the inverse, to check calculations.

Subtraction question 1

3241 people visited the zoo on Saturday.

On Sunday 2876 people visited.

How many more people visited the lake on Saturday than on Sunday?

Answer (1 mark): 365

 3241 – 2876 = 365

Subtraction question 2

 A teacher prints out 1242 worksheets in a term.

If 435 were maths worksheets, how many did she print out for the other subjects?

Answer (1 mark): 807

Subtraction question 3

The temperature in Toronto dropped to minus 15 degrees celcius in December. 

In July the temperature was 47 degrees celsius warmer than it was in December. What was the temperature in July? 

Answer (1 mark): 32 degrees warmer

47 – 15 degrees = 32 degrees celsius

Counting on 47 degrees from minus 15 degrees  = 32 degrees celsius

In Year 4, multiplication word problems can include recalling facts for times tables up to 12 x 12 and multiplying two and three-digit numbers by a 1-digit number, using formal written layout.  

Multiplication question 1

All the pupils in Year 4 complete a mental maths test.

27 pupils score 9 marks out of 10.

What is the total number of marks scored by the 27 pupils?

Answer (1 mark): 243

27 x 9 = 243

Multiplication question 2

Year 3 and 4 children from a local primary school go on a school trip. 

Six mini buses are used to transport the children.

 There are 17 children on each minibus.

How many children go on the school trip?

Answer (1 mark): 102

17 x 6 = 102 children

Multiplication question 3

Biscuits come in packs of 18.

Mrs Smith buys 8 packs for the parents at the Y4 maths workshop.

How many biscuits does she buy altogether?

Answer (1 mark): 144 biscuits

Division word problems in year 4 require pupils to be able to recall division facts for multiplication tables up to 12 x 12. Formal written method of division isn’t a requirement until Year 5 however, many schools choose to teach the formal method in Year 4. Pupils need to understand the concept of grouping and sharing and to understand the link between multiplication & division.

Division question 1

Sam has 28 friends coming to his birthday party.

Each child will receive a cupcake, which come in packs of 4.

How many packs of cupcakes will Sam need to buy?

Answer (1 mark): 7 packs

28 ÷ 4 = 7 

Division question 2

4 children raised £96 between them on a sponsored walk.

If they split the money evenly between the four of them, how much did each pupil raise?

Answer (1 mark): £24 each

96 ÷ 4 = 24

Division question 3

Ahmed is thinking of a number

He says, ‘when I divide my number by 12, the answer is 108.

What number was Ahmed thinking of?

Answer (1 mark): 9

108 ÷ 12 = 9

In Year 4, decimal and fraction problems involve increasingly harder fractions to calculate quantities and fractions to divide quantities, including non-unit fractions, where the answer is a whole number. Decimal word problems include measure and money problems involving fractions and decimals up to 2 decimal places.

Fraction and decimal question 1

Jamie has 18 sweets. 

He gives \frac{1}{6} of the sweets to his friend and keeps the rest himself.

How many sweets does Jamie have now?

Answer (1 mark): 15 sweets

\frac{1}{6} of 18 = 3

18 – 3 = 15

Fraction and decimal question 2

Jaxon collected 36 conkers. 

\frac{1}{4} of the conkers fell out of a hole in his bag, when he was walking home.

How many conkers did Jaxon have left, when he got home?

\frac{1}{4} of 36 = 9

36 – 9 = 37      or      \frac{3}{4} of 36 = 29 (3 x 9)

Fraction and decimal question 3

Sara ate \frac{3}{12} of a chocolate bar and gave \frac{2}{12} to her friend.

What fraction of the chocolate bar did she have left? 

 Answer (1 mark): \frac{7}{12}

In Year 4, time word problems include: converting from hours to minutes, minutes to seconds, years to months and weeks to days.

Time question 1 

A cake was put in the oven at 4:35pm and taken out at 4:57pm.

How long was the cake in the oven?

Answer (1 mark): 22 minutes

57-35 = 22 minutes

Using an number line: 

Time question 2

It took Evie 25 minutes to complete a page of number problems. 

If she started at 2:45pm. What time did she finish?

Answer (1 mark): 3:10pm

In Year 4, children are introduced to multi-step word problems requiring up to two steps. These problems cover a range of concepts, including the four operations, fractions, decimals and measures.

Third Space Learning’s online one-to-one tutoring frequently incorporates multi-step questions to test students’ knowledge and problem solving skills. Our personalised tutoring programme works to identify gaps in students’ learning, fill those gaps, reinforce students’ knowledge and build confidence.

Multi-step question 1

There are 6 handwriting pens in each pack.

A class has 30 children and each child needs 2 handwriting pens.

How many packs will the teacher need to buy?

Answer (2 marks): 10 packs of handwriting pens.

30 x 2 = 60

Multi-step question 2

 Sophie has £4.50.

She buy 3 books at a carboot sale, costing 50p, 65p and £1.20.

How much money does she have left?

Answer (2 marks): £2.15 left

Multi-step question 3

Abullah is thinking of a number.

He doubles the number and adds 7.

He gets an answer of 25. 

What was his original number?

Answer (2 marks): 9

Third Space Learning offers word problems for all primary year groups. Take a look at our word problems for year 3 , word problems for year 5 and word problems for year 6 . Our word problems span a range of topics such as ratio word problems and percentage word problems .

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

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Learn how tutors develop pupils’ maths fluency or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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Resources tagged with: NC Yr 4

There are 51 NRICH Mathematical resources connected to NC Yr 4 , you may find related items under NC .

Satisfying Four Statements

Can you find any two-digit numbers that satisfy all of these statements?

Representing Numbers

Find as many different ways of representing this number of dots as you can.

Ordering Journeys

How would you put these journey lengths in order?

What Distance?

Can you use addition and subtraction to answer these questions about real-life distances?

Count Me In

How do you know whether you will reach these numbers when you count in steps of six from zero?

Dicey Operations in Line

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Seeing Squares

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Reasoned Rounding

Four strategy dice games to consolidate pupils' understanding of rounding.

Round the Dice Decimals 1

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Let Us Divide!

Look at different ways of dividing things. What do they mean? How might you show them in a picture, with things, with numbers and symbols?

Discuss and Choose

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

Bryony's Triangle

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Sorting Logic Blocks

This activity focuses on similarities and differences between shapes.

Light the Lights Again

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

A task which depends on members of the group noticing the needs of others and responding.

What Shape?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Counters in the Middle

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Table Patterns Go Wild!

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

Times Tables Shifts

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Reflector ! Rotcelfer

Can you place the blocks so that you see the reflection in the picture?

Nice or Nasty

There are nasty versions of this dice game but we'll start with the nice ones...

The Remainders Game

Play this game and see if you can figure out the computer's chosen number.

Four-digit Targets

You have two sets of the digits 0-9. Can you arrange these in the five boxes to make four-digit numbers as close to the target numbers as possible?

Venn Diagrams

How will you complete these interactive Venn diagrams?

Eight Hidden Squares

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Shape Times Shape

These eleven shapes each stand for a different number. Can you use the number sentences to work out what they are?

Multiplication Square Jigsaw

Can you complete this jigsaw of the multiplication square?

Twice as Big?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Multiples Grid

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Coordinate Challenge

Use the clues about the symmetrical properties of these letters to place them on the grid.

Torn Shapes

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Take Your Dog for a Walk

Use the interactivity to move Pat. Can you reproduce the graphs and tell their story?

Fractional Wall

Using the picture of the fraction wall, can you find equivalent fractions?

Stringy Quads

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Nine-pin Triangles

How many different triangles can you make on a circular pegboard that has nine pegs?

Carrying Cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Andy's Marbles

Andy had a big bag of marbles but unfortunately the bottom of it split and all the marbles spilled out. Use the information to find out how many there were in the bag originally.

How Big Are Classes 5, 6 and 7?

Use the two sets of data to find out how many children there are in Classes 5, 6 and 7.

Fractional Triangles

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

The Deca Tree

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

Symmetry Challenge

How many symmetric designs can you make on this grid? Can you find them all?

Let Us Reflect

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Fractions in a Box

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

Shapes on the Playground

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

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• Published: 22 April 2024

The design and evaluation of gamified online role-play as a telehealth training strategy in dental education: an explanatory sequential mixed-methods study

• Chayanid Teerawongpairoj 1 ,
• Chanita Tantipoj 1 &
• Kawin Sipiyaruk 2  

Scientific Reports volume  14 , Article number:  9216 ( 2024 ) Cite this article

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Metrics details

• Health care
• Health services
• Public health

To evaluate user perceptions and educational impact of gamified online role-play in teledentistry as well as to construct a conceptual framework highlighting how to design this interactive learning strategy, this research employed an explanatory sequential mixed-methods design. Participants were requested to complete self-perceived assessments toward confidence and awareness in teledentistry before and after participating in a gamified online role-play. They were also asked to complete a satisfaction questionnaire and participate in an in-depth interview to investigate their learning experience. The data were analyzed using descriptive statistics, paired sample t-test, one-way analysis of variance, and framework analysis. There were 18 participants who completed self-perceived assessments and satisfaction questionnaire, in which 12 of them participated in a semi-structured interview. There were statistically significant increases in self-perceived confidence and awareness after participating in the gamified online role-play ( P  < 0.001). In addition, the participants were likely to be satisfied with this learning strategy, where usefulness was perceived as the most positive aspect with a score of 4.44 out of 5, followed by ease of use (4.40) and enjoyment (4.03). The conceptual framework constructed from the qualitative findings has revealed five key elements in designing a gamified online role-play, including learner profile, learning settings, pedagogical components, interactive functions, and educational impact. The gamified online role-play has demonstrated its potential in improving self-perceived confidence and awareness in teledentistry. The conceptual framework developed in this research could be considered to design and implement a gamified online role-play in dental education. This research provides valuable evidence on the educational impact of gamified online role-play in teledentistry and how it could be designed and implemented in dental education. This information would be supportive for dental instructors or educators who are considering to implement teledentistry training in their practice.

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Introduction.

Telehealth has gained significant attention from various organization due to its potential to improve healthcare quality and accessibility 1 . It can be supportive in several aspects in healthcare, including medical and nursing services, to enhance continuous monitoring and follow-up 2 . Its adoption has increased substantially during the COVID-19 pandemic, aiming to provide convenient healthcare services 3 . Even though the COVID-19 outbreak has passed, many patients still perceive telehealth as an effective tool in reducing a number of visits and enhancing access to health care services 4 , 5 . This supports the use of telehealth in the post-COVID-19 era.

Teledentistry, a form of telehealth specific to dentistry, has been employed to improve access to dental services 6 . This system offers benefits ranging from online history taking, oral diagnosis, treatment monitoring, and interdisciplinary communication among dental professionals, enabling comprehensive and holistic treatment planning for patients 7 . Teledentistry can also reduce travel time and costs associated with dental appointments 8 , 9 , 10 . There is evidence that teledentistry serves as a valuable tool to enhance access to dental care for patients 11 . Additionally, in the context of long-term management in patients, telehealth has contributed to patient-centered care, by enhancing their surrounding environments 12 . Therefore, teledentistry should be emphasized as one of digital dentistry to enhance treatment quality.

Albeit the benefits of teledentistry, available evidence demonstrates challenges and concerns in the implementation of telehealth. Lack of awareness and knowledge in the use of telehealth can hinder the adoption of telehealth 13 . Legal issues and privacy concerns also emerge as significant challenges in telehealth use 14 . Moreover, online communication skills and technology literacy, including competency in using technological tools and applications, have been frequently reported as challenges in teledentistry 15 , 16 . Concerns regarding limitations stemming from the lack of physical examination are also significant 17 . These challenges and complexities may impact the accuracy of diagnosis and the security and confidentiality of patient information. Therefore, telehealth training for dental professionals emerges as essential prerequisites to effectively navigate the use of teledentistry, fostering confidence and competence in remote oral healthcare delivery.

The feasibility and practicality of telehealth in dental education present ongoing challenges and concerns. Given the limitations of teledentistry compared to face-to-face appointments, areas of training should encompass the telehealth system, online communication, technical issues, confidentiality concerns, and legal compliance 18 . However, there is currently no educational strategy that effectively demonstrates the importance and application of teledentistry 19 . A role-play can be considered as a teaching strategy where learners play a role that closely resembles real-life scenarios. A well-organized storytelling allows learner to manage problematic situations, leading to the development of problem-solving skill 20 , 21 . When compared to traditional lecture-based learning, learners can also enhance their communication skills through conversations with simulated patients 22 , 23 . In addition, they could express their thoughts and emotions during a role-play through experiential learning 20 , 24 , 25 . Role-play through video teleconference would be considered as a distance learning tool for training dental professionals to effectively use teledentistry.

While there have been studies supporting online role-play as an effective learning tool due to its impact of flexibility, engagement, and anonymity 26 , 27 , no evidence has been yet reported whether or not this learning strategy could have potential for training teledentistry. Given the complicated issues in telehealth, role-play for training teledentistry should incorporate different learning aspects compared to face-to-face communication with patients. In addition, game components have proved to be supportive in dental education 28 , 29 . Consequently, this research aimed to evaluate user perceptions and educational impact of gamified online role-play to enhance learner competence and awareness in using teledentistry as well as to construct a conceptual framework highlighting how to design and implement this interactive learning strategy. This research would introduce and promote the design and implementation of gamified online role-play as a learning tool for training teledentistry. To achieve the aim, specific objectives were established as follows:

1. To design a gamified online role-play for teledentistry training.

2. To investigate learner perceptions regarding their confidence and awareness in the use of teledentistry after completing the gamified online role-play.

3. To explore user satisfactions toward the use of gamified online role-play.

4. To develop a conceptual framework for designing and implementing a gamified online role-play for teledentistry training.

Materials and methods

Research design.

This research employed an explanatory sequential mixed-methods design, where a quantitative phase was firstly performed followed by a qualitative phase 30 , 31 . The quantitative phase was conducted based on pre-experimental research using one-group pretest–posttest design. Participants were requested to complete self-perceived assessments toward confidence and awareness in the use of teledentistry before and after participating in a gamified online role-play. They were also asked to complete a satisfaction questionnaire in using a gamified online role-play for training teledentistry. The qualitative phase was afterwards conducted to explore in-depth information through semi-structured interviews, in order to enhance an understanding of the quantitative phase, and to develop a conceptual framework for designing and implementing an online role-play for training teledentistry.

A gamified online role-play for training teledentistry

A gamified online role-play was designed and developed by the author team. To ensure its educational impact was significant, the expected learning outcomes were formulated based on insights gathered from a survey with experienced instructors from the Department of Advanced General Dentistry, Faculty of Dentistry, Mahidol University. These learning outcomes covered areas of online communication skill, technical issues, technology literacy of patients, limitations of physical examination, and privacy concerns of personal information. Learning scenario and instructional content were subsequently designed to support learners in achieving the expected learning outcomes, with their alignments validated by three experts in dental education. A professional actress underwent training to role-play a patient with a dental problem, requesting a virtual consultation or teledentistry. Before conducting data collection, the simulated patient was required to undergo a training and adjusting process with a pilot group under supervision of two experts in advanced general dentistry and dental education who had experience with teledentistry to ensure realism and completeness of learning content.

According to the role-play scenario, an actress was assigned to portray a 34-year-old female with chief complaints of pain around both ears, accompanied by difficulties in chewing food due to tooth loss. She was instructed to express her anxiety and nervousness about addressing these issues. Additionally, it was specified that she could not take a day off from work during this period. Despite this constraint, she required a dental consultation to receive advice for initial self-care, as her symptoms significantly impacted her daily life. Furthermore, she was designated to encounter difficulties with the technological use of the teledentistry platform.

The game components were implemented into the online role-play to enhance motivation and engagement. As challenge and randomness appear to be game elements 32 , 33 , five challenge cards were designed and embedded into the online role-play, where a participant was asked to randomly select one of them before interacting with the simulated patient. The challenging situations were potential technical concerns which could occur frequently during video conferencing, including network problems (e.g., internet disconnection and poor connection) and audiovisual quality issues. The participants were blinded to the selected card, while it was revealed to only the simulated patient. The challenging conditions were mimicked by the organizers and simulated patient, allowing learners to deal with difficulties. Therefore, both challenges and randomness were implemented into this learning intervention not only to create learning situations but also to enhance engagement.

A feedback system was carefully considered and implemented into the gamified online role-play. Immediate feedback appears to be a key feature of interactive learning environments 29 . Formative feedback was instantly delivered to learners through verbal and non-verbal communication, including words (content), tone of voice, facial expressions, and gestures of the simulated patient. This type of feedback allowed participants to reflect on whether or not their inputs were appropriate, enabling them to learn from their mistakes, or so-called the role of failure 34 . Summative feedback was also provided at the end of the role-play through a reflection from a simulated patient and suggestions from an instructor.

Learners were able to interact with the simulated patient using an online meeting room by Cisco WebEx. According to the research setting (Fig.  1 ), a learner was asked to participate in the role-play activity using a computer laptop in a soundproof room, while a simulated patient was arranged in a prepared location showing her residential environment. The researcher and instructor also joined the online meeting room and observed the interaction between the simulated patient and learners during the role-play activity whether or not all necessary information was accurately obtained. The role-play activity took around 30 minutes.

A diagram demonstrating the setting of gamified online role-play.

Research participants

Quantitative phase.

The participants in this research were postgraduate students from the Residency Training Program in Advanced General Dentistry at Mahidol University Faculty of Dentistry in academic year 2022, using a volunteer sampling. This program was selected because its objective was to develop graduates capable of integrating competencies from various dental disciplines to provide comprehensive dental care for both normal patients and those with special needs. Therefore, teledentistry should be a supportive component of their service. The recruitment procedure involved posting a recruiting text in the group chat of the residents. Those interested in participating in the research were informed to directly contact us to request more information, and they were subsequently allowed to decide whether they would like to participate. This approach ensured that participation was voluntary. Although there could be a non-response bias within this non-probability sampling technique 35 , it was considered as appropriate for this study, as participants were willing to have contribution in the learning activity, and therefore accurate and reliable research findings with no dropout could be achieved 36 .

The inclusion and exclusion criteria were established to determine the eligibility of prospective participants for this research. This study included postgraduate students from Years 1 to 3 in the Residency Training Program in Advanced General Dentistry at Mahidol University Faculty of Dentistry, enrolled during the academic year 2022. They were also required to at least complete the first semester to be eligible for this research to ensure familiarity with comprehensive dental care. However, they were excluded if they had previous involvement in the pilot testing of the gamified online role-play or if they were not fluent in the Thai language. The sample size was determined using a formula for two dependent samples (comparing means) 37 . To detect a difference in self-perceived confidence and awareness between pre- and post-assessments at a power of 90% and a level of statistical significance of 1%, five participants were required. With an assumed dropout rate of 20%, the number of residents per year (Year 1–3) was set to be 6. Therefore, 18 residents were required for this research.

Qualitative phase

The participants from the quantitative phase were selected for semi-structured interviews using a purposive sampling. This sampling method involved the selection of information-rich participants based on specific criteria deemed relevant to the research objective and to ensure a diverse representation of perspectives and experiences within the sample group 38 . In this research, the information considered for the purposive sampling included demographic data (e.g., sex and year of study), along with self-perceived assessment scores. By incorporating perceptions from a variety of participants, a broad spectrum of insights from different experiences in comprehensive dental practice and diverse improvement levels in self-perceived confidence and awareness could inform the design and implementation of the training program effectively. The sample size for this phase was determined based on data saturation, wherein interviews continued until no new information or emerging themes were retrieved. This method ensured thorough exploration of the research topic and maximized the richness of the qualitative data obtained.

Outcome assessments

To evaluate the gamified online role-play, a triangular design approach was employed, enabling the researchers to compare the research outcomes from different assessment methods. In this research, self-perceived assessments (confidence and awareness) in teledentistry, satisfactions toward gamified online role-play, and learner experience were assessed to assure the quality and feasibility of the gamified online role-play.

Self-perceived confidence and awareness toward teledentistry

All participants were requested to rate their perceptions of teledentistry before and after participating in the gamified online role-play (Supplementary material 1 ). The self-perceived assessment was developed based on previous literature 39 , 40 , 41 , 42 . The assessment scores would inform whether or not the participants could improve their self-perceived confidence and awareness through a learning activity. The assessment consisted of two parts, which were (1) self-perceived confidence and (2) self-perceived awareness. Each part contained six items, which were similar between the pre- and post-assessments. All items were designed using a 5-point Likert scale, where 1 being ‘strongly disagree’ and 5 being ‘strongly agree’.

Satisfactions toward the gamified online role-play

All participants were asked to complete the satisfaction questionnaire after participating in the gamified online role-play, to investigate whether or not they felt satisfied with their learning (Supplementary material 2 ). The questionnaire was developed based on previous literature regarding gamification and role-play 41 , 42 , 43 , 44 . Most of the items were designed using a 5-point Likert scale, where 1 being ‘very dissatisfied’ and 5 being ‘very satisfied’. They were grouped into three aspects, which were (1) Perceived usefulness, (2) Perceived ease of use, and (3) Perceived enjoyment.

Learner experiences within the gamified online role-play

Semi-structured interviews were conducted with the purposively selected participants to gather in-depth information regarding their learning experiences within the gamified online role-play. This technique allowed researchers to ask additional interesting topics raised from the responses of participants. A topic guide for interviews were constructed based on the findings of previous literature 45 , 46 , 47 . The interview was conducted in a private room by a researcher who was trained in conducting qualitative research including interviews. The interview sessions took approximately 45–60 minutes, where all responses from participants were recorded using a digital audio recorder with their permission. The recorded audios were transcribed using a verbatim technique by a transcription service under a confidential agreement.

Validity and reliability of data collection tools

To enhance the quality of self-perceived assessment and satisfaction questionnaire, they were piloted and revised to assure their validity and reliability. According to the content validity, three experts in advanced general dentistry were asked to evaluate the questionnaire, where problematic items were iteratively revised until they achieved the index of item-objective congruence (IOC) higher than 0.5. To perform a test–retest reliability, the validated versions of both self-perceived assessment and satisfaction questionnaire were afterwards piloted in residents from other programs, and the data were analyzed using an intraclass correlation coefficient (ICC), where the values of all items were 0.7 or greater. The data from the first pilot completion of both data collection tools were analyzed using Cronbach’s alpha to ensure the internal consistency of all constructs. The problematic items were deleted to achieve the coefficient alpha of 0.7 or greater for all constructs, which was considered as acceptable internal consistency.

Data analysis

The quantitative data retrieved from self-perceived assessment and satisfaction questionnaire were analyzed with the Statistical Package for Social Sciences software (SPSS, version 29, IBM Corp.). Descriptive statistics were performed to present an overview of the data. The scores from pre- and post-assessments were analyzed using a paired sample t-test to evaluate whether or not the participants would better self-perceive their confidence and awareness in teledentistry after participating in the gamified online role-play. One-way analysis of variance (ANOVA) was conducted to compare whether or not there were statistically significant differences in self-perceived assessment and satisfaction scores among the three academic years.

The qualitative data retrieved from semi-structured interviews were analyzed using a framework analysis, where its procedure involved transcription, familiarization with the interview data, coding, developing an analytical framework, indexing, charting, and data interpreting qualitative findings 48 . In this research, the initial codes had been pre-defined from previous literature and subsequently adjusted following the analysis of each transcript to develop an analytical framework (themes and subthemes), requiring several iterations until no additional codes emerged. Subsequently, the established categories and codes were applied consistently across all transcripts (indexing). The data from each transcript were then charted to develop a matrix, facilitating the management and summarization of qualitative findings. This method enabled the researchers to compare and contrast differences within the data and to identify connections between categories, thereby exploring their relationships and informing data interpretation.

The procedure of framework analysis necessitated a transparent process for data management and interpretation of emerging themes to ensure the robustness of research 49 . The transparency of this analytic approach enabled two researchers (C.Te. and K.S.) to independently analyze the qualitative data, and the emerging themes afterwards were discussed to obtain consensus among the researchers. This technique can be considered as a triangular approach to assure the intercoder reliability and internal validity of this research. The transparent process also allowed an external expert in dental education to verify the accuracy of the analysis. All emerging themes and the decision on data saturation were based on a discussion of all researchers until an agreement was made. NVivo (version 14, QSR International) was used to performed the qualitative data analysis. Subsequently, a conceptual framework was constructed to demonstrate emerging themes and subthemes together with their relationships.

Ethical consideration

The ethical approval for the study was approved by the Institutional Review Board of Faculty of Dentistry and Faculty of Pharmacy, Mahidol University on 29 th September 2022, the ethical approval number: MU-DT/PY-IRB 2022/049.2909. All methods were performed in accordance with the relevant guidelines and regulations. Although the data were not anonymous in nature as they contained identifiable data, they were coded prior to the analysis to assure confidentiality of participants.

Informed consent

Informed consent was obtained from all participants.

There were 18 residents from Year 1 to 3 of the Residency Training Program in Advanced General Dentistry who participated in this research (six from each year). Of these, there were 14 females and 4 males. There was no participant dropout, as all of them completed all required tasks, including the pre- and post-perceived assessments, gamified online role-play, and satisfaction questionnaire. According to the purposive sampling, the participants from the quantitative phase were selected for semi-structured interviews by considering sex, year of study, and self-perceived assessment scores. Twelve students (ten females and two males) participated in semi-structured interviews, where their characteristics are presented in Table 1 .

Internal consistency of all constructs

The data collected from the research participants, in addition to the pilot samples, were analyzed with Cronbach’s alpha to confirm the internal consistency. The coefficient alpha of all constructs demonstrated high internal consistency, as demonstrated in Table 2 .

Self-perceived assessments toward confidence and awareness of teledentistry

There were statistically significant increases in the assessment scores of self-perceived confidence and awareness after participating in the gamified online role-play ( P  < 0.001). According to Table 3 , there was an increase in self-perceived confidence from 3.38 (SD = 0.68) for the pre-assessment to 4.22 (SD = 0.59) for the post-assessment ( P  < 0.001). The findings of self-perceived awareness also showed score improvement from 4.16 (SD = 0.48) to 4.55 (SD = 0.38) after interacting with the simulated patient ( P  < 0.001).

According to Fig.  2 , participants demonstrated a higher level of self-perceived assessments for both self-confidence and awareness in all aspects after participating in the gamified online role-play for teledentistry training.

Self-perceived assessments toward confidence and awareness of teledentistry.

When comparing the self-perceived assessment scores toward confidence and awareness in the use of teledentistry among the three years of study (Year 1–3), there were no statistically significant differences in the pre-assessment, post-assessment score, and score difference (Table 4 ).

Satisfactions toward the use of gamified online role-play

According to Fig.  3 , participants exhibited high levels of satisfaction with the use of gamified online role-play across all three aspects. The aspect of usefulness received the highest satisfaction rating with a score of 4.44 (SD = 0.23) out of 5, followed by ease of use and enjoyment, scoring 4.40 (SD = 0.23) and 4.03 (SD = 0.21), respectively. Particularly, participants expressed the highest satisfaction levels regarding the usefulness of gamified online role-play for identifying their role (Mean = 4.72, SD = 0.46) and developing problem-solving skills associated with teledentistry (Mean = 4.61, SD = 0.50). Additionally, they reported satisfaction with the learning sequence presented in the gamified online role-play (Mean = 4.61, SD = 0.50). However, participants did not strongly perceive that the format of the gamified online role-play could engage them with the learning task for an extended period (Mean = 3.72, SD = 0.83).

Satisfactions toward the use of gamified online role-play.

When comparing the satisfaction levels perceived by participants from different academic years (Table 5 ), no statistically significant differences were observed among the three groups for all three aspects ( P  > 0.05).

Following the framework analysis of qualitative data, there were five emerging themes, including: (1) learner profile, (2) learning settings of the gamified online role-play, (3) pedagogical components, (4) interactive functions, and (5) educational impact.

Theme 1: Learner profile

Learner experience and preferences appeared to have impact on how the participants perceived the use of gamified online role-play for teledentistry training. When learners preferred role-play or realized benefits of teledentistry, they were likely to support this learning intervention. In addition, they could have seen an overall picture of the assigned tasks before participating in this research.

“I had experience with a role-play activity when I was dental undergraduates, and I like this kind of learning where someone role-plays a patient with specific personalities in various contexts. This could be a reason why I felt interested to participate in this task (the gamified online role-play). I also believed that it would be supportive for my clinical practice.” Participant 12, Year 1, Female “Actually, I' have seen in several videos (about teledentistry), where dentists were teaching patients to perform self-examinations, such as checking their own mouth and taking pictures for consultations. Therefore, I could have thought about what I would experience during the activity (within the gamified online role-play).” Participant 8, Year 2, Female

Theme 2: Learning settings of the gamified online role-play

Subtheme 2.1: location.

Participants had agreed that the location for conducting a gamified online role-play should be in a private room without any disturbances, enabling learners to focus on the simulated patient. This could allow them to effectively communicate and understand of the needs of patient, leading to a better grasp of lesson content. In addition, the environments of both learners and simulated patient should be authentic to the learning quality.

“The room should be a private space without any disturbances. This will make us feel confident and engage in conversations with the simulated patient.” Participant 10, Year 1, Female “… simulating a realistic environment can engage me to interact with the simulated patient more effectively ...” Participant 8, Year 2, Female

Subtheme 2.2: Time allocated for the gamified online role-play

The time allocated for the gamified online role-play in this research was considered as appropriate, as participants believed that a 30-minutes period should be suitable to take information and afterwards give some advice to their patient. In addition, a 10-minutes discussion on how they interact with the patient could be supportive for participants to enhance their competencies in the use of teledentistry.

“… it would probably take about 20 minutes because we would need to gather a lot of information … it might need some time to request and gather various information … maybe another 10-15 minutes to provide some advice.” Participant 7, Year 1, Female “I think during the class … we could allocate around 30 minutes for role-play, … we may have discussion of learner performance for 10-15 minutes ... I think it should not be longer than 45 minutes in total.” Participant 6, Year 2, Female

Subtheme 2.3: Learning consequence within a postgraduate curriculum

Most participants suggested that the gamified online role-play in teledentistry should be arranged in the first year of their postgraduate program. This could maximize the effectiveness of online role-play, as they would be able to implement teledentistry for their clinical practice since the beginning of their training. However, some participants suggested that this learning approach could be rearranged in either second or third year of the program. As they already had experience in clinical practice, the gamified online role-play would reinforce their competence in teledentistry.

"Actually, it would be great if this session could be scheduled in the first year … I would feel more comfortable when dealing with my patients through an online platform." Participant 11, Year 2, Male "I believe this approach should be implemented in the first year because it allows students to be trained in teledentistry before being exposed to real patients. However, if this approach is implemented in either the second or third year when they have already had experience in patient care, they would be able to better learn from conversations with simulated patients." Participant 4, Year 3, Male

Theme 3: Pedagogical components

Subtheme 3.1: learning content.

Learning content appeared to be an important component of pedagogical aspect, as it would inform what participants should learn from the gamified online role-play. Based on the interview data, participants reported they could learn how to use a video teleconference platform for teledentistry. The conditions of simulated patient embedded in an online role-play also allowed them to realize the advantages of teledentistry. In addition, dental problems assigned to the simulated patient could reveal the limitations of teledentistry for participants.

“The learning tasks (within the gamified online role-play) let me know how to manage patients through the teleconference.” Participant 5, Year 2, Female “… there seemed to be limitations (of teledentistry) … there could be a risk of misdiagnosis … the poor quality of video may lead to diagnostic errors … it is difficult for patients to capture their oral lesions.” Participant 3, Year 2, Female

Subtheme 3.2: Feedback

During the use of online role-play, the simulated patient can provide formative feedback to participants through facial expressions and tones of voice, enabling participants to observe and learn to adjust their inquiries more accurately. In addition, at the completion of the gamified online role-play, summative feedback provided by instructors could summarize the performance of participants leading to further improvements in the implementation of teledentistry.

“I knew (whether or not I interacted correctly) from the gestures and emotions of the simulated patient between the conversation. I could have learnt from feedback provided during the role-play, especially from the facial expressions of the patient.” Participant 11, Year 2, Male “The feedback provided at the end let me know how well I performed within the learning tasks.” Participant 2, Year 1, Female

Theme 4: Interactive functions

Subtheme 4.1: the authenticity of the simulated patient.

Most participants believed that a simulated patient with high acting performance could enhance the flow of role-play, allowing learners to experience real consequences. The appropriate level of authenticity could engage learners with the learning activity, as they would have less awareness of time passing in the state of flow. Therefore, they could learn better from the gamified online role-play.

"It was so realistic. ... This allowed me to talk with the simulated patient naturally ... At first, when we were talking, I was not sure how I should perform … but afterwards I no longer had any doubts and felt like I wanted to explain things to her even more." Participant 3, Year 2, Female "At first, I believed that if there was a factor that could influence learning, it would probably be a simulated patient. I was impressed by how this simulated patient could perform very well. It made the conversation flow smoothly and gradually." Participant 9, Year 3, Female

Subtheme 4.2: Entertaining features

Participants were likely to be satisfied with the entertaining features embedded in the gamified online role-play. They felt excited when they were being exposed to the unrevealed challenge which they had randomly selected. In addition, participants suggested to have more learning scenarios or simulated patients where they could randomly select to enhance randomness and excitement.

“It was a playful experience while communicating with the simulated patient. There are elements of surprise from the challenge cards that make the conversation more engaging, and I did not feel bored during the role-play.” Participant 4, Year 3, Male “I like the challenge card we randomly selected, as we had no idea what we would encounter … more scenarios like eight choices and we can randomly choose to be more excited. I think we do not need additional challenge cards, as some of them have already been embedded in patient conditions.” Participant 5, Year 2, Female

Subtheme 4.3: Level of difficulty

Participants suggested the gamified online role-play to have various levels of difficulty, so learners could have a chance to select a suitable level for their competence. The difficulties could be represented through patient conditions (e.g., systemic diseases or socioeconomic status), personal health literacy, and emotional tendencies. They also recommended to design the gamified online role-play to have different levels where learners could select an option that is suitable for them.

“The patient had hidden their information, and I needed to bring them out from the conversation.” Participant 12, Year 1, Female “Patients' emotions could be more sensitive to increase level of challenges. This can provide us with more opportunities to enhance our management skills in handling patient emotions.” Participant 11, Year 2, Male “… we can gradually increase the difficult level, similar to playing a game. These challenges could be related to the simulated patient, such as limited knowledge or difficulties in communication, which is likely to occur in our profession.” Participant 6, Year 2, Female

Theme 5: Educational impact

Subtheme 5.1: self-perceived confidence in teledentistry, communication skills.

Participants were likely to perceive that they could learn from the gamified online role-play and felt more confident in the use of teledentistry. This educational impact was mostly achieved from the online conversation within the role-play activity, where the participants could improve their communication skills through a video teleconference platform.

“I feel like the online role-play was a unique form of learning. I believe that I gained confidence from the online communication the simulated patient. I could develop skills to communicate effectively with real patients.” Participant 11, Year 2, Male “I believe it support us to train communication skills ... It allowed us to practice both listening and speaking skills more comprehensively.” Participant 4, Year 3, Male

Critical thinking and problem-solving skills

In addition to communication skills, participants reported that challenges embedded in the role-play allowed them to enhance critical thinking and problem-solving skills, which were a set of skills required to deal with potential problems in the use of teledentistry.

"It was a way of training before experiencing real situations … It allowed us to think critically whether or not what we performed with the simulated patients was appropriate." Participant 7, Year 1, Female “It allowed us to learn how to effectively solve the arranged problems in simulated situation. We needed to solve problems in order to gather required information from the patient and think about how to deliver dental advice through teledentistry.” Participant 11, Year 2, Male

Subtheme 5.2: Self perceived awareness in teledentistry

Participants believed that they could realize the necessity of teledentistry from the gamified online role-play. The storytelling or patient conditions allowed learners to understand how teledentistry could have both physical and psychological support for dental patients.

“From the activity, I would consider teledentistry as a convenient tool for communicating with patients, especially if a patient cannot go to a dental office”. Participant 5, Year 2, Female “I learned about the benefits of teledentistry, particularly in terms of follow-up. The video conference platform could support information sharing, such as drawing images or presenting treatment plans, to patients.” Participant 8, Year 2, Female

A conceptual framework of learning experience within a gamified online role-play

Based on the qualitative findings, a conceptual framework was developed in which a gamified online role-play was conceptualized as a learning strategy in supporting learners to be able to implement teledentistry in their clinical practice (Fig.  4 ).

The conceptual framework of key elements in designing a gamified online role-play.

The conceptual framework has revealed key elements to be considered in designing a gamified online role-play. Learner profile, learning settings, pedagogical components, and interactive functions are considered as influential factors toward user experience within the gamified online role-play. The well-designed learning activity will support learners to achieve expected learning outcomes, considered as educational impact of the gamified online role-play. The contributions of these five key elements to the design of gamified online role-play were interpreted, as follows:

Learner profile: This element tailors the design of gamified online role-plays for teledentistry training involves considering the background knowledge, skills, and experiences of target learners to ensure relevance and engagement.

Learning settings: The element focuses the planning for gamified online role-plays in teledentistry training involves selecting appropriate contexts, such as location and timing, to enhance accessibility and achieve learning outcomes effectively.

Pedagogical components: This element emphasizes the alignment between learning components and learning outcomes within gamified online role-plays, to ensure that the content together with effective feedback design can support learners in improving their competencies from their mistakes.

Interactive functions: This element highlights interactivity features integrated into gamified online role-plays, such as the authenticity and entertaining components to enhance immersion and engagement, together with game difficulty for optimal flow. All these features should engage learners with the learning activities until the achievement of learner outcomes.

Educational impact: This element represents the expected learning outcomes, which will inform the design of learning content and activities within gamified online role-plays. In addition, this element could be considered to evaluate the efficacy of gamified online role-plays, reflecting how well learning designs align with the learning outcomes.

A gamified online role-play can be considered as a learning strategy for teledentistry according to its educational impact. This pedagogical approach could mimic real-life practice, where dental learners could gain experience in the use of teledentistry in simulated situations before interacting with actual patients. Role-play could provide learners opportunities to develop their required competencies, especially communication and real-time decision-making skills, in a predictable and safe learning environment 20 , 23 , 46 . Potential obstacles could also be arranged for learners to deal with, leading to the enhancement of problem-solving skill 50 . In addition, the recognition of teledentistry benefits can enhance awareness and encourage its adoption and implementation, which could be explained by the technology acceptance model 51 . Therefore, a gamified online role-play with a robust design and implementation appeared to have potential in enhancing self-perceived confidence and awareness in the use of teledentistry.

The pedagogical components comprised learning content, which was complemented by assessment and feedback. Learners could develop their competence with engagement through the learning content, gamified by storytelling of the online role-play 52 , 53 . Immediate feedback provided through facial expression and voice tone of simulated patients allowed participants to learn from their failure, considered as a key feature of game-based learning 29 , 45 . The discussion of summative feedback provided from an instructor at the end of role-play activity could support a debriefing process enabling participants to reflect their learning experience, considered as important of simulation-based game 54 . These key considerations should be initially considered in the design of gamified online role-play.

The interactive functions can be considered as another key component for designing and evaluating the gamified online role-play 45 . Several participants enjoyed with a learning process within the gamified online role-play and suggested it to have more learning scenarios. In other words, this tool could engage learners with an instructional process, leading to the achievement of learning outcomes 29 , 45 . As challenge and randomness appear to be game elements 32 , 33 , this learning intervention assigned a set of cards with obstacle tasks for learners to randomly pick up before interacting with simulated patients, which was perceived by participants as a feature to make the role-play more challenging and engaging. This is consistent with previous research, where challenging content for simulated patients could make learners more engaged with a learning process 55 . However, the balance between task challenges and learner competencies is certainly required for the design of learning activities 56 , 57 . The authenticity of simulated patient and immediate feedback could also affect the game flow, leading to the enhancement of learner engagement 45 . These elements could engage participants with a learning process, leading to the enhancement of educational impact.

The educational settings for implementing gamified online role-play into dental curriculum should be another concern. This aspect has been recognized as significant in existing evidence 45 . As this research found no significant differences in all aspects among the three groups of learners, this learning intervention demonstrated the potential for its implementation at any time of postgraduate dental curriculum. This argument can be supported by previous evidence where a role-play could be adaptable for learning at any time, as it requires a short learning period but provides learners with valuable experience prior to being exposed in real-life scenarios 58 . This strategy also provides opportunities for learners who have any question or concern to seek advice or guidance from their instructors 59 . Although the gamified online role-play can be arranged in the program at any time, the first academic year should be considered, as dental learners would be confidence in implementing teledentistry for their clinical practice.

While a gamified online role-play demonstrated its strengths as an interactive learning strategy specifically for teledentistry, there are a couple of potential drawbacks that need to be addressed. The requirement for synchronous participation could limit the flexibility of access time for learners (synchronous interactivity limitation). With only one learner able to engage with a simulated patient at a time (limited participants), more simulated patients would be required if there are a number of learners, otherwise they would need to wait for their turn. Time and resources are significantly required for preparing simulated patients 60 . Despite the use of trained and calibrated professional actors/actresses, inauthenticity may be perceived during role-plays, requiring a significant amount of effort to achieve both interactional and clinical authenticities 46 . Future research could investigate asynchronous learning approaches utilizing non-player character (NPC) controlled by an artificial intelligence system as a simulated patient. This setup would enable multiple learners to have the flexibility to engage with the material at their own pace and at times convenient to them 29 . While there are potential concerns about using gamified online role-plays, this interactive learning intervention offers opportunities for dental professionals to enhance their teledentistry competency in a safe and engaging environment.

Albeit the robust design and data collection tools to assure reliability and validity as well as transparency of this study, a few limitations were raised leading to a potential of further research. While this research recruited only postgraduate students to evaluate the feasibility of gamified online role-play in teledentistry training, further research should include not only experienced dental practitioners but also undergraduate students to confirm its potential use in participants with different learner profiles. More learning scenarios in other dental specialties should also be included to validate its effectiveness, as different specialties could have different limitations and variations. Additional learning scenarios from various dental disciplines should be considered to validate the effectiveness of gamified online role-plays, as different specialties may present unique limitations and variations. A randomized controlled trial with robust design should be required to compare the effectiveness of gamified online role-play with different approaches in training the use of teledentistry.

Conclusions

This research supports the design and implementation of a gamified online role-play in dental education, as dental learners could develop self-perceived confidence and awareness with satisfaction. A well-designed gamified online role-play is necessary to support learners to achieve expected learning outcomes, and the conceptual framework developed in this research can serve as a guidance to design and implement this interactive learning strategy in dental education. However, further research with robust design should be required to validate and ensure the educational impact of gamified online role-play in dental education. Additionally, efforts should be made to develop gamified online role-play in asynchronous learning approaches to enhance the flexibility of learning activities.

Data availability

The data that support the findings of this study are available from the corresponding author, up-on reasonable request. The data are not publicly available due to information that could compromise the privacy of research participants.

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Acknowledgements

The authors would like to express our sincere gratitude to participants for their contributions in this research. We would also like to thank the experts who provided their helpful suggestions in the validation process of the data collection tools.

This research project was funded by the Faculty of Dentistry, Mahidol University. The APC was funded by Mahidol University.

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Teerawongpairoj, C., Tantipoj, C. & Sipiyaruk, K. The design and evaluation of gamified online role-play as a telehealth training strategy in dental education: an explanatory sequential mixed-methods study. Sci Rep 14 , 9216 (2024). https://doi.org/10.1038/s41598-024-58425-9

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Modified crayfish optimization algorithm for solving multiple engineering application problems

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• Published: 24 April 2024
• Volume 57 , article number  127 , ( 2024 )

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• Heming Jia 1 ,
• Xuelian Zhou 1 ,
• Jinrui Zhang 1 ,
• Laith Abualigah 2 , 3 ,
• Ali Riza Yildiz 4 &
• Abdelazim G. Hussien 5  

Crayfish Optimization Algorithm (COA) is innovative and easy to implement, but the crayfish search efficiency decreases in the later stage of the algorithm, and the algorithm is easy to fall into local optimum. To solve these problems, this paper proposes an modified crayfish optimization algorithm (MCOA). Based on the survival habits of crayfish, MCOA proposes an environmental renewal mechanism that uses water quality factors to guide crayfish to seek a better environment. In addition, integrating a learning strategy based on ghost antagonism into MCOA enhances its ability to evade local optimality. To evaluate the performance of MCOA, tests were performed using the IEEE CEC2020 benchmark function and experiments were conducted using four constraint engineering problems and feature selection problems. For constrained engineering problems, MCOA is improved by 11.16%, 1.46%, 0.08% and 0.24%, respectively, compared with COA. For feature selection problems, the average fitness value and accuracy are improved by 55.23% and 10.85%, respectively. MCOA shows better optimization performance in solving complex spatial and practical application problems. The combination of the environment updating mechanism and the learning strategy based on ghost antagonism significantly improves the performance of MCOA. This discovery has important implications for the development of the field of optimization.

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1 Introduction

For a considerable period, engineering application problems have been widely discussed by people. At present, improving the modern scientific level of engineering construction has become the goal of human continuous struggle, including constrained engineering design problems (Zhang et al. 2022a ; Mortazavi 2019 ) affected by a series of external factors and feature selection problems (Kira and Rendell 1992 ), and so on. Constrained engineering design problems refers to the problem of achieving optimization objectives and reducing calculation costs under many external constraints, which is widely used in mechanical engineering (Abualigah et al. 2022 ), electrical engineering (Razmjooy et al. 2021 ), civil engineering (Kaveh 2017 ), chemical engineering (Talatahari et al. 2021 ) and other engineering fields, such as workshop scheduling (Meloni et al. 2004 ), wind power generation (Lu et al. 2021 ), and UAV path planning (Belge et al. 2022 ), parameter extraction of photovoltaic models(Zhang et al. 2022b ; Zhao et al. 2022 ), Optimization of seismic foundation isolation system (Kandemir and Mortazavi 2022 ), optimal design of RC support foundation system of industrial buildings (Kamal et al. 2023 ), synchronous optimization of fuel type and external wall insulation performance of intelligent residential buildings (Moloodpoor and Mortazavi 2022 ), economic optimization of double-tube heaters (Moloodpoor et al. 2021 ).

Feature selection is the process of choosing specific subsets of features from a larger set based on defined criteria. In this approach, each original feature within the subset is individually evaluated using an assessment function. The aim is to select pertinent features that carry distinctive characteristics. This selection process reduces the dimensionality of the feature space, enhancing the model's generalization ability and accuracy. The ultimate goal is to create the best possible combination of features for the model. By employing feature selection, the influence of irrelevant factors is minimized. This reduction in irrelevant features not only streamlines the computational complexity but also reduces the time costs associated with processing the data. Through this method, redundant and irrelevant features are systematically removed from the model. This refinement improves the model’s accuracy and results in a higher degree of fit, ensuring that the model aligns more closely with the underlying data patterns.

In practical applications of feature selections, models are primarily refined using two main methods: the filter (Cherrington et al. 2019 ) and wrapper (Jović et al. 2015 ) techniques. The filter method employs a scoring mechanism to assess and rank the model's features. It selects the subset of features with the highest scores, considering it as the optimal feature combination. On the other hand, the wrapper method integrates the selection process directly into the learning algorithm. It embeds the feature subset evaluation within the learning process, assessing the correlation between the chosen features and the model. In recent years, applications inspired by heuristic algorithms can be seen everywhere in our lives and are closely related to the rapid development of today's society. These algorithms play an indispensable role in solving a myriad of complex engineering problems and feature selection challenges. They have proven particularly effective in addressing spatial, dynamic, and random problems, showcasing significant practical impact and tangible outcomes.

With the rapid development of society and science and technology, through continuous exploitation and exploration in the field of science, more and more complex and difficult to describe multi-dimensional engineering problems also appear in our research process. Navigating these complexities demands profound contemplation and exploration. While traditional heuristic algorithms have proven effective in simpler, foundational problems, they fall short when addressing the novel and intricate multi-dimensional challenges posed by our current scientific landscape and societal needs. Thus, researchers have embarked on a journey of continuous contemplation and experimentation. By cross-combining and validating existing heuristic algorithms, they have ingeniously devised a groundbreaking solution: Metaheuristic Algorithms (MAs) (Yang 2011 ). This innovative approach aims to tackle the complexities of our evolving problems, ensuring alignment with the rapid pace of social and technological development. MAs is a heuristic function based algorithm. It works by evaluating the current state of the problem and possible solutions to guide the algorithm in making choices in the search space. MAs improves the efficiency and accuracy of the problem solving process by combining multiple heuristic functions and updating the search direction at each step based on their weights. The diversity of MAs makes it a universal problem solver, adapting to the unique challenges presented by different problem domains. Essentially represents a powerful paradigm shift in computational problem solving, providing a powerful approach to address the complexity of modern engineering and scientific challenges. Compared with traditional algorithms, MAs has made great progress in finding optimal solutions, jumping out of local optima, and overcoming convergence difficulties in the later stage of solution through the synergy of different algorithms. These enhancements mark a significant progress, which not only demonstrates the adaptability of the scientific method, but also emphasizes the importance of continuous research and cooperation. It also has the potential to radically solve problems in domains of complex engineering challenges, enabling researchers to navigate complex problem landscapes with greater accuracy and efficiency.

Research shows that MAs are broadly classified into four different research directions: swarm-based, natural evolution-based, human-based, and physics-based. These categories include a wide range of innovative problem-solving approaches, each drawing inspiration from a different aspect of nature, human behavior, or physical principles. Researchers exploration these different pathways to solve complex challenges and optimize the solutions efficiently. First of all, the swarm-based optimization algorithm is the optimization algorithm that uses the wisdom of population survival to solve the problem. For example, Particle Swarm Optimization Algorithm (PSO) (Wang et al. 2018a ) is an optimization algorithm based on the group behavior of birds. PSO has a fast search speed and is only used for real-valued processing. However, it is not good at handling discrete optimization problems and has fallen into local optimization. Artificial Bee Colony Optimization Algorithm (ABC) (Jacob and Darney 2021 ) realizes the sharing and communication of information among individuals when bees collect honey according to their respective division of labor. In the Salp Swarm Algorithm (SSA) (Mirjalili et al. 2017 ), individual sea squirts are connected end to end and move and prey in a chain, and follow the leader with followers according to a strict “hierarchical” system. Ant Colony Optimization Algorithm (ACO) (Dorigo et al. 2006 ), ant foraging relies on the accumulation of pheromone on the path, and spontaneously finds the optimal path in an organized manner.

Secondly, a natural evolutionary algorithm inspired by the law of group survival of the fittest, an optimization algorithm that finds the best solution by preserving the characteristics of easy survival and strong individuals, such as: Genetic Programming Algorithm (GP) (Espejo et al. 2009 ), because biological survival and reproduction have certain natural laws, according to the structure of the tree to deduce certain laws of biological genetic and evolutionary process. Evolutionary Strategy Algorithm (ES) (Beyer and Schwefel 2002 ), the ability of a species to evolve itself to adapt to the environment, and produce similar but different offspring after mutation and recombination from the parent. Differential Evolution (DE) (Storn and Price 1997 ) eliminates the poor individuals and retains the good ones in the process of evolution, so that the good ones are constantly approaching the optimal solution. It has a strong global search ability in the initial iteration, but when there are fewer individuals in the population, individuals are difficult to update, and it is easy to fall into the local optimal. The Biogeography-based Optimization Algorithm (BBO) (Simon 2008 ), influenced by biogeography, filters out the global optimal value through the iteration of the migration and mutation of species information.

Then, Human-based optimization algorithms are optimization algorithms that take advantage of the diverse and complex human social relationships and activities in a specific environment to solve problems, such as: The teaching–learning-based Optimization (TLBO) (Rao and Rao 2016 ) obtained the optimal solution by simulating the Teaching relationship between students and teachers. It simplifies the information sharing mechanism within each round, and all evolved individuals can converge to the global optimal solution faster, but the algorithm often loses its advantage when solving some optimization problems far from the origin. Coronavirus Mask Protection Algorithm (CMPA) (Yuan et al. 2023 ), which is mainly inspired by the self-protection process of human against coronavirus, establishes a mathematical model of self-protection behavior and solves the optimization problem. Cultural Evolution Algorithm (CEA) (Kuo and Lin 2013 ), using the cultural model of system thinking framework for exploitation to achieve the purpose of cultural transformation, get the optimal solution. Volleyball Premier League Algorithm (VPL) (Moghdani and Salimifard 2018 ) simulates the process of training, competition and interaction of each team in the volleyball game to solve the global optimization problem.

Finally, Physics-based optimization algorithm is an optimization algorithm that uses the basic principles of physics to simulate the physical characteristics of particles in space to solve problems. For example, Snow Ablation Algorithm (SAO) (Deng and Liu 2023 ), inspired by the physical reaction of snow in nature, realizes the transformation among snow, water and steam by simulating the sublation and ablation of snow. RIME Algorithm (RIME) (Su et al. 2023 ) is a exploration and exploitation of mathematical model balance algorithm based on the growth process of soft rime and hard rime in nature. Central Force Optimization Algorithm (CFO) (Formato 2007 ), aiming at the problem of complex calculation of the initial detector, a mathematical model of uniform design is proposed to reduce the calculation time. Sine and cosine algorithm (SCA) (Mirjalili 2016 ) establishes mathematical models and seeks optimal solutions based on the volatility and periodicity characteristics of sine and cosine functions. Compared with the candidate solution set of a certain scale, the algorithm has a strong search ability and the ability to jump out of the local optimal, but the results of some test functions fluctuate around the optimal solution, and there is a certain precocious situation, and the convergence needs to be improved.

While the original algorithm is proposed, many improved MAs algorithms are also proposed to further improve the optimization performance of the algorithm in practical application problems, such as: Yujun-Zhang et al. combined the arithmetic optimization algorithm (AOA) with the Aquila Optimizer(AO) algorithm to propose a new meta-heuristic algorithm (AOAAO) (Zhang et al. 2022c ). CSCAHHO algorithm (Zhang et al. 2022d ) is a new algorithm obtained by chaotic mixing of sine and cosine algorithm (SCA) and Harris Hqwk optimization algorithm (HHO). Based on LMRAOA algorithm proposed to solve numerical and engineering problems (Zhang et al. 2022e ). Yunpeng Ma et al. proposed an improved teaching-based optimization algorithm to artificially reduce NOx emission concentration in circulating fluidized bed boilers (Ma et al. 2021 ). The improved algorithm SOS(MSOS) (Kumar et al. 2019 ), based on the natural Symbiotic search (SOS) algorithm, improves the search efficiency of the algorithm by introducing adaptive return factors and modified parasitic vectors. Modified beluga whale optimization with multi-strategies for solving engineering problems (MBWO) (Jia et al. 2023a ) by gathering Beluga populations for feeding and finding new habitats during long-distance migration. Betul Sultan Yh-ld-z et al. proposed a novel hybrid optimizer named AO-NM, which aims to optimize engineering design and manufacturing problems (Yıldız et al. 2023 ).

The Crayfish Optimization Algorithm (COA) (Jia et al. 2023b ) is a novel metaheuristic algorithm rooted in the concept of population survival wisdom, introduced by Heming Jia et al. in 2023. Drawing inspiration from crayfish behavior, including heat avoidance, competition for caves, and foraging, COA employs a dual-stage strategy. During the exploration stage, it replicates crayfish searching for caves in space for shelter, while the exploitation stage mimics their competition for caves and search for food. Crayfish, naturally averse to dry heat, thrive in freshwater habitats. To simulate their behavior and address challenges related to high temperatures and food scarcity, COA incorporates temperature variations into its simulation. By replicating crayfish habits, the algorithm dynamically adapts to environmental factors, ensuring robust problem-solving capabilities. Based on temperature fluctuations, crayfish autonomously select activities such as seeking shelter, competing for caves, and foraging. When the temperature exceeds 30°C, crayfish instinctively seek refuge in cool, damp caves to escape the heat. If another crayfish is already present in the cave, a competition ensues for occupancy. Conversely, when the temperature drops below 30°C, crayfish enter the foraging stage. During this phase, they make decisions about food consumption based on the size of the available food items. COA achieves algorithmic transformation between exploration and exploitation stages by leveraging temperature variations, aiming to balance the exploration and exploitation capabilities of the algorithm. However, COA solely emulates the impact of temperature on crayfish behavior, overlooking other significant crayfish habits, leading to inherent limitations. In the latter stages of global search, crayfish might cluster around local optimum positions, restricting movement. This hampers the crayfish's search behavior, slowing down convergence speed, and increasing the risk of falling into local optima, thereby making it challenging to find the optimal solution.

In response to the aforementioned challenges, this paper proposes a Modified Crayfish Optimization Algorithm (MCOA). MCOA introduces an environmental update mechanism inspired by crayfish's preference for living in fresh flowing water. MCOA incorporates crayfish's innate perception abilities to assess the quality of the surrounding aquatic environment, determining whether the current habitat is suitable for survival. The simulation of crayfish crawling upstream to find a more suitable aquatic environment is achieved by utilizing adaptive flow factors and leveraging the crayfish's second, third foot perceptions to determine the direction of water flow.This method partially replicates the survival and reproduction behavior of crayfish, ensuring the continual movement of the population. It heightens the randomness within the group, widens the search scope for crayfish, enhances the algorithm's exploration efficiency, and effectively strengthens the algorithm’s global optimization capabilities. Additionally, the ghost opposition-based learning strategy (Jia et al. 2023c ) is implemented to introduce random population initialization when the algorithm becomes trapped in local optima. This enhancement significantly improves the algorithm's capability to escape local optima, promoting better exploration of the solution space. After the careful integration of the aforementioned two strategies, the search efficiency and predation speed of the crayfish algorithm experience a substantial improvement. Moreover, the algorithm's convergence rate and global optimization ability are significantly enhanced, leading to more effective and efficient problem-solving capabilities.

In the experimental section, we conducted a comprehensive comparison between MCOA and nine other metaheuristic algorithms. We utilized the IEEE CEC2020 benchmark function to evaluate the performance of the algorithm. The evaluation involved statistical methods such as the Wilcoxon rank sum test and Friedman test to rank the averages, validating the efficiency of the MCOA algorithm and the effectiveness of the proposed improvements. Furthermore, MCOA was applied to address four constrained engineering design problems as well as the high-dimensional feature selection problem using the wrapper method. These practical applications demonstrated the practicality and effectiveness of MCOA in solving real-world engineering problems.

The main contributions of this paper are as follows:

In the environmental renewal mechanism, the water quality factor and roulette wheel selection method are introduced to simulate the process of crayfish searching for a more suitable water environment for survival.

The introduction of the ghost opposition-based learning strategy enhances the randomness of crayfish update locations, effectively preventing the algorithm from getting trapped in local optima, and improving the overall global optimization performance of the algorithm.

The fixed value of food intake is adaptively adjusted based on the number of evaluations, enhancing the algorithm's capacity to escape local optima. This adaptive change ensures a more dynamic exploration of the solution space, improving the algorithm's overall optimization effectiveness.

The MCOA’s performance is compared with nine metaheuristics, including COA, using the IEEE CEC2020 benchmark function. The comparison employs the Wilcoxon rank sum test and Friedman test to rank the averages, providing evidence for the efficiency of MCOA and the effectiveness of the proposed improvements.

The application of MCOA to address four constrained engineering design problems and the high-dimensional feature selection problem using the wrapper method demonstrates the practicality and effectiveness of MCOA in real-world applications.

The main structure of this paper is as follows, the first part of the paper serves as a brief introduction to the entire document, providing an overview of the topics and themes that will be covered. In the second part, the paper provides a comprehensive summary of the Crayfish Optimization Algorithm (COA). In the third part, a modified crawfish optimization algorithm (MCOA) is proposed. By adding environment updating mechanism and ghost opposition-based learning strategy, MCOA can enhance the global search ability and convergence speed to some extent. Section four shows the experimental results and analysis of MCOA in IEEE CEC2020 benchmark functions. The fifth part applies MCOA to four kinds of constrained engineering design problems. In Section six, MCOA is applied to the high-dimensional feature selection problem of wrapper methods to demonstrate the effectiveness of MCOA in practical application problems. Finally, Section seven concludes the paper.

2 Crayfish optimization algorithm (COA)

Crayfish is a kind of crustaceans living in fresh water, its scientific name is crayfish, also called red crayfish or freshwater crayfish, because of its food, fast growth rate, rapid migration, strong adaptability and the formation of absolute advantages in the ecological environment. Changes in temperature often cause changes in crayfish behavior. When the temperature is too high, crayfish choose to enter the cave to avoid the damage of high temperature, and when the temperature is suitable, they will choose to climb out of the cave to forage. According to the living habits of crayfish, it is proposed that the three stages of summer, competition for caves and going out to forage correspond to the three living habits of crayfish, respectively.

Crayfish belong to ectotherms and are affected by temperature to produce behavioral differences, which range from 20 °C to 35 °C. The temperature is calculated as follows:

where temp represents the temperature of the crayfish's environment.

2.1 Initializing the population

In the d -dimensional optimization problem of COA, each crayfish is a 1 ×  d matrix representing the solution of the problem. In a set of variables ( X 1 , X 2 , X 3 …… X d ), the position ( X ) of each crayfish is between the upper boundary ( ub ) and lower boundary ( lb ) of the search space. In each evaluation of the algorithm, an optimal solution is calculated, and the solutions calculated in each evaluation are compared, and the optimal solution is found and stored as the optimal solution of the whole problem. The position to initialize the crayfish population is calculated using the following formula.

where X i,j denotes the position of the i-th crayfish in the j-th dimension, ub j denotes the upper bound of the j-th dimension, lb j denotes the lower bound of the j-th dimension, and rand is a random number from 0 to 1.

2.2 Summer escape stage (exploration stage)

In this paper, the temperature of 30 °C is assumed to be the dividing line to judge whether the current living environment is in a high temperature environment. When the temperature is greater than 30 ℃ and it is in the summer, in order to avoid the harm caused by the high temperature environment, crayfish will look for a cool and moist cave and enter the summer to avoid the influence of high temperature. The caverns are calculated as follows.

where X G represents the optimal position obtained so far for this evaluation number, and X L represents the optimal position of the current population.

The behavior of crayfish competing for the cave is a random event. To simulate the random event of crayfish competing for the cave, a random number rand is defined, when rand < 0.5 means that there are no other crayfish currently competing for the cave, and the crayfish will go straight into the cave for the summer. At this point, the crayfish position update calculation formula is as follows.

Here, X new is the next generation position after location update, and C 2 is a decreasing curve. C 2 is calculated as follows.

Here, FEs represents the number of evaluations and MaxFEs represents the maximum number of evaluations.

2.3 Competition stage (exploitation stage)

When the temperature is greater than 30 °C and rand ≥ 0.5, it indicates that the crayfish have other crayfish competing with them for the cave when they search for the cave for summer. At this point, the two crayfish will struggle against the cave, and crayfish X i adjusts its position according to the position of the other crayfish X z . The adjustment position is calculated as follows.

Here, z represents the random individual of the crayfish, and the random individual calculation formula is as follows.

where, N is the population size.

2.4 Foraging stage (exploitation stage)

The foraging behavior of crayfish is affected by temperature, and temperature less than or equal to 30 ℃ is an important condition for crayfish to climb out of the cave to find food. When the temperature is less than or equal to 30 °C, the crayfish will drill out of the cave and judge the location of the food according to the optimal location obtained in this evaluation, so as to find the food to complete the foraging. The position of the food is calculated as follows.

The amount of food crayfish eat depends on the temperature. When the temperature is between 20 °C and 30°C, crayfish have strong foraging behavior, and the most food is found and the maximum food intake is also obtained at 25 °C. Thus, the food intake pattern of crayfish resembles a normal distribution. Food intake was calculated as follows.

Here, µ is the most suitable temperature for crayfish feeding, and σ and C 1 are the parameters used to control the variation of crayfish intake at different temperatures.

The food crayfish get depends not only on the amount of food they eat, but also on the size of the food. If the food is too large, the crayfish can't eat the food directly. They need to tear it up with their claws before eating the food. The size of the food is calculated as follows.

Here, C 3 is the food factor, which represents the largest food, and its value is 3, fitness i represents the fitness value of the i-th crayfish, and fitness food represents the fitness value of the location of the food.

Crayfish use the value of the maximum food Q to judge the size of the food obtained and thus decide the feeding method. When Q  > ( C 3  + 1)/2, it means that the food is too large for the crayfish to eat directly, and it needs to tear the food with its claws and eat alternately with the second and third legs. The formula for shredding food is as follows.

After the food is shredded into a size that is easy to eat, the second and third claws are used to pick up the food and put it into the mouth alternately. In order to simulate the process of bipedal eating, the mathematical models of sine function and cosine function are used to simulate the crayfish eating alternately. The formula for crayfish alternating feeding is as follows.

When Q  ≤ ( C 3  + 1)/2, it indicates that the food size is suitable for the crayfish to eat directly at this time, and the crayfish will directly move towards the food location and eat directly. The formula for direct crayfish feeding is as follows.

2.5 Pseudo-code for COA

Crayfish optimization algorithm pseudo-code

3 Modified crayfish optimization algorithm (MCOA)

Based on crayfish optimization algorithm, we propose a high-dimensional feature selection problem solving algorithm (MCOA) based on improved crayfish optimization algorithm. In MCOA, we know that the quality of the aquatic environment has a great impact on the survival of crayfish, according to the living habits of crayfish, which mostly feed on plants and like fresh water. Oxygen is an indispensable energy for all plants and animals to maintain life, the higher the content of dissolved oxygen in the water body, the more vigorous the feeding of crayfish, the faster the growth, the less disease, and the faster the water flow in the place of better oxygen permeability, more aquatic plants, suitable for survival, so crayfish has a strong hydrotaxis. When crayfish perceive that the current environment is too dry and hot or lack of food, they crawl backward according to their second, third and foot perception (r) to judge the direction of water flow, and find an aquatic environment with sufficient oxygen and food to sustain life. Good aquatic environment has sufficient oxygen and abundant aquatic plants, to a certain extent, to ensure the survival and reproduction of crayfish.

In addition, we introduce ghost opposition-based learning to help MCOA escape the local optimal trap. The ghost opposition-based learning strategy combines the candidate individual, the current individual and the optimal individual to randomly generate a new candidate position to replace the previous poor candidate position, and then takes the best point or the candidate solution as the central point, and then carries out more specific and extensive exploration of other positions. Traditional opposition-based learning (Mahdavi et al. 2018 ) is based on the central point and carries out opposition-based learning in a fixed format. Most of the points gather near the central point and their positions will not exceed the distance between the current point and the central point, and most solutions will be close to the optimal individual. However, if the optimal individual is not near the current exploration point, the algorithm will fall into local optimal and it is difficult to find the optimal solution. Compared with traditional opposition-based learning, ghost opposition-based learning is a opposition-based learning solution that can be dynamically changed by adjusting the size of parameter k, thereby expanding the algorithm's exploration range of space, effectively solving the problem that the optimal solution is not within the search range based on the center point, and making the algorithm easy to jump out of the local optimal.

According to the life habits of crayfish, this paper proposes a Modified Crayfish Optimization Algorithm (MCOA), which uses environment update mechanism and ghost opposition-based learning strategy to improve COA, and shows the implementation steps, pseudo-code and flow chart of MCOA algorithm as follows.

3.1 Environment update mechanism

In the environmental renewal mechanism, a water quality factor V is introduced to represent the quality of the aquatic environment at the current location. In order to simplify the design and computational complexity of the system, the water quality factor V of the MCOA is represented by a hierarchical discretization, and its value range is set to 0 to 5. Crayfish perceive the quality of the current aquatic environment through the perception ( r ) of the second and third legs, judge whether the current living environment can continue to survive through the perception, and independently choose whether to update the current location. The location update is calculated as follows.

Among them, each crayfish has a certain difference in its own perception of water environment r , X 2 is a random position between the candidate optimal position and the current position, which is calculated by Eq. ( 15 ), X 1 is a random position in the population, and B is an adaptive water flow factor, which is calculated by Eq. ( 16 ).

Among them, the sensing force r of the crayfish’s second and third legs is a random number [0,1]. c is a constant that represents the water flow velocity factor with a value of 2. When V  ≤ 3, it indicates that the crayfish perceives the quality of the current living environment to be good and is suitable for continued survival. When V > 3, it indicates that the crayfish perceives that the current living environment quality is poor, and it needs to crawl in the opposite direction according to the direction of water flow that crayfish perceives, so as to find an aquatic environment with sufficient oxygen and abundant food Fig.  1 .

Classification of MAs

In the environmental updating mechanism, in order to describe the behavior of crayfish upstream in more detail, the perception area of crayfish itself is abstractly defined as a circle in MCOA, and crayfish is in the center of the circle. In each evaluation calculation, a random Angle θ is first calculated by the roulette wheel selection algorithm to determine the moving direction of the crayfish in the circular area, and then the moving path of the crayfish is determined according to the current moving direction. In the whole circle, random angles can be chosen from 0 to 360 degrees, from which the value of θ can be determined to be of magnitude [− 1,1]. The difference of random Angle indicates that each crayfish moves its position in a random direction, which broadens the search range of crayfish, enhances the randomness of position and the ability to escape from local optimum, and avoids local convergence Fig.  2 .

Schematic diagram of the environment update mechanism

3.2 Ghost opposition-based learning strategy

The ghost opposition-based learning strategy takes a two-dimensional space as an example. It is assumed that there is a two-dimensional space, as shown in Fig.  3 . On the X-axis, [ ub , lb ] represents the search range of the solution, and the ghost generation method is shown in Fig.  3 . Assuming that the position of a new candidate solution is Xnew and the height of the solution is h1 i , the position of the best solution on the X-axis is the projected position of the candidate solution, and the position and height are XG , h2 i , respectively. In addition, on the X-axis there is a projection position X i of the candidate solution with a height of h3 i. Thus, the position of the ghost is obtained. The projection position of the ghost on the X-axis is x i by vector calculation, and its height is h i . The ghost position is calculated using the following formula.

Schematic diagram of ghost opposition-based learning strategy

In Fig.  3 , the Y-axis represents the convex lens. Suppose there is a ghost position P i , where x i is its projection on the X-axis and h i is its height. P* i is the real image obtained by convex lens imaging. P* i is projected on the X-axis as x* i and has height h* i . Therefore, the opposite individual x* i of individual x i can be obtained. x* i is the corresponding point corresponding to the ghost individual x i obtained from O as the base point. According to the lens imaging principle, we can obtain Eq. ( 18 ), and the calculation formula is as follows.

The strategy formula of ghost opposition-based learning is evolved from Eq. ( 18 ). The strategy formula of ghost opposition-based learning is calculated as follows.

3.3 Implementation of MCOA algorithm

3.3.1 initialization phase.

Initialize the population size N , the population dimension d , and the number of evaluations FEs . The initialized population is shown in Eq. ( 2 ).

3.3.2 Environment update mechanism

Crayfish judge the quality of the current aquatic environment according to the water quality factor V , and speculate whether the current aquatic environment can continue to survive. When V  > 3 indicates that the crawfish perceives the quality of the current aquatic environment as poor and is not suitable for survival. According to the sensory information of the second and third legs and the adaptive flow factor, the crawfish judges the direction of the current flow, and then moves upstream to find a better aquatic environment to update the current position. The position update formula is shown in Eq. ( 14 ). When V  < 3, it means that the crayfish has a good perception of the current living environment and is suitable for survival, and does not need to update its position.

3.3.3 Exploration phase

When the temperature is greater than 30 ℃ and V  > 3, it indicates that crayfish perceive the current aquatic environment quality is poor, and the cave is dry and without moisture, which cannot achieve the effect of summer vacation. It is necessary to first update the position by crawling in the reverse direction according to the flow direction, and find a cool and moist cave in a better quality aquatic environment for summer.

3.3.4 Exploitation stage

When the temperature is less than 30 ℃ and V  > 3, it indicates that crayfish perceive the current aquatic environment is poor, and there is not enough food to maintain the survival of crayfish. It is necessary to escape from the current food shortage living environment by crawling in the reverse direction according to the current direction, and find a better aquatic environment to maintain the survival and reproduction of crayfish.

3.3.5 Ghost opposition-based learning strategy

Through the combination of the candidate individual, the current individual and the optimal individual, a candidate solution is randomly generated and compared with the current solution, the better individual solution is retained, the opposite individual is obtained, and the location of the ghost is obtained. The combination of multiple positions effectively prevents the algorithm from falling into local optimum, and the specific implementation formula is shown in Eq. ( 19 ).

3.3.6 Update the location

The position of the update is determined by comparing the fitness values. If the fitness of the current individual update is better, the current individual replaces the original individual. If the fitness of the original individual is better, the original individual is retained to exist as the optimal solution.

The pseudocode for MCOA is as follows (Algorithm 2).

Modified Crayfish optimization algorithm pseudo-code

The flow chart of the MCOA algorithm is as follows.

3.4 Computational complexity analysis

The complexity analysis of algorithms is an essential step to evaluate the performance of algorithms. In the experiment of complexity analysis of the algorithm, we choose the IEEE CEC2020 Special Session and Competition as the complexity evaluation standard of the single objective optimization algorithm. The complexity of MCOA algorithm mainly depends on several important parameters, such as the population size ( N  = 30), the number of dimensions of the problem ( d  = 10), the maximum number of evaluations of the algorithm ( MaxFEs  = 100,000) and the solution function ( C ). Firstly, the running time of the test program is calculated and the running time ( T 0 ) of the test program is recorded, and the test program is shown in Algorithm 3. Secondly, under the same dimension of calculating the running time of the test program, the 10 test functions in the IEEE CEC2020 function set were evaluated 100,000 times, and their running time ( T 1 ) was recorded. Finally, the running time of 100,000 evaluations of 10 test functions performed by MCOA for 5 times under the same dimension was recorded, and the average value was taken as the running time of the algorithm ( T 2 ). Therefore, the formula for calculating the time complexity of MCOA algorithm is given in Eq. ( 21 ).

IEEE CEC2020 complexity analysis test program

The experimental data table of algorithm complexity analysis is shown in Table  1 . In the complexity analysis of the algorithm, we use the method of comparing MCOA algorithm with other seven metaheuristic algorithms to illustrate the complexity of MCOA. In Table  1 , we can see that the complexity of MCOA is much lower than other comparison algorithms such as ROA, STOA, and AOA. However, compared with COA, the complexity of MCOA is slightly higher than that of COA because it takes a certain amount of time to update the location through the environment update mechanism and ghost opposition-based learning strategy. Although the improved strategy of MCOA increases the computation time to a certain extent, the optimization performance of MOCA has been significantly improved through a variety of experiments in section four of this paper, which proves the good effect of the improved strategy.

4 Experimental results and discussion

The experiments are carried out on a 2.50 GHz 11th Gen Intel(R) Core(TM) i7-11,700 CPU with 16 GB memory and 64-bit Windows11 operating system using Matlab R2021a. In order to verify the performance of MCOA algorithm, MCOA is compared with nine metaheuristic algorithms in this subsection. In the experiments, we used the IEEE CEC2020 test function to evaluate the optimization performance of the MCOA algorithm Fig.  4 .

Flow chart of the MCOA algorithm

4.1 Experiments with IEEE CEC2020 test functions

In this subsection, using the Crayfish Optimization Algorithm (COA), Remora Optimization Algorithm (ROA) (Jia et al. 2021 ), Sooty Tern Optimization Algorithm (STOA) (Dhiman and Kaur 2019 ), Arithmetic Optimization Algorithm (AOA) (Abualigah et al. 2021 ), Harris Hawk Optimization Algorithm (HHO) (Heidari et al. 2019 ), Prairie Dog Optimization Algorithm (PDO) (Ezugwu et al. 2022 ), Genetic Algorithm (GA) (Mirjalili and Mirjalili 2019 ),Modified Sand Cat Swarm Optimization Algorithm (MSCSO) (Wu et al. 2022 ) and a competition algorithm LSHADE (Piotrowski 2018 ) were compared to verify the optimization effect of MCOA. The parameter Settings of each algorithm are shown in Table  2 .

In order to test the performance of MCOA, this paper selects 10 benchmark test functions of IEEE CEC2020 for simulation experiments. Where F1 is a unimodal function, F2–F3 is a multimodal function, F4 is a non-peak function, F5–F7 is a hybrid function, and F8-F10 is a composite function. The parameters of this experiment are uniformly set as follows: the maximum number of evaluation MaxFEs is 100,000, the population size N is 30, and the dimension size d is 10. The MCOA algorithm and the other nine algorithms are run independently for 30 times, and the average fitness value, standard deviation of fitness value and Friedman ranking calculation of each algorithm are obtained. The specific function Settings of the IEEE CEC2020 benchmark functions are shown in Table  3 .

4.1.1 Results statistics and convergence curve analysis of IEEE CEC2020 benchmark functions

In order to more clearly and intuitively compare the ability of MCOA and various algorithms to find individual optimal solutions, the average fitness value, standard deviation of fitness value and Friedman ranking obtained by running MCOA and other comparison algorithms independently for 30 times are presented in the form of tables and images. The data and images are shown in Table  4 and Fig.  5 respectively.

Convergence curve of MCOA algorithm in IEEE CEC2020

In Table  4 , mean represents the average fitness value, std represents the standard deviation of fitness value, rank represents the Friedman ranking, Friedman average rank represents the average ranking of the algorithm among all functions, and Friedman rank represents the final ranking of this algorithm. Compared with other algorithms, MCOA achieved the best results in average fitness value, standard deviation of fitness value and Friedman ranking. In unimodal function F1, although MCOA algorithm is slightly worse than LSHADE algorithm, MCOA is superior to other algorithms in mean fitness value, standard deviation of fitness value, Friedman ranking and other aspects. In the multimodal functions F2 and F3, although the average fitness value of MCOA is slightly worse, it also achieves a good result of ranking second. The standard deviation of fitness value in F3 is better than other comparison algorithms in terms of stability. In the peakless function F4, except GA and LSHADE algorithm, other algorithms can find the optimal individual solution stably. In the mixed functions F5, F6, and F7, although the mean fitness value of LSHADE is better than that of MCOA, the standard deviation of the fitness value of MCOA is better than that of the other algorithms compared. Among the composite functions of F8, F9 and F10, the standard deviation of MCOA's fitness value at F8 is slightly worse than that of LSHADE, but the average fitness value and standard deviation of fitness value are the best in other composite functions, and it has achieved the first place in all composite functions. Finally, from the perspective of Friedman average rank, MCOA has a strong comprehensive performance and still ranks first. Through the analysis of the data in Table  4 , it can be seen that MCOA ranks first overall and has good optimization effect, and its optimization performance is better than other 9 comparison algorithms.

Figure  5 shows that in the IEEE CEC2020 benchmark functions, for the unimodal function F1, although LSHADE algorithm has a better optimization effect, compared with similar meta-heuristic algorithms, MCOA has a slower convergence rate in the early stage, but can be separated from local optimal and converge quickly in the middle stage. In the multimodal functions F2 and F3, similar to F1, MCOA converges faster in the middle and late stages, effectively exiting the local optimal. Although the convergence speed is slower than that of LSHADE, the optimal value can still be found. In the peak-free function F4, the optimal value can be found faster by all algorithms except LSHADE, STOA and PDO because the function is easy to implement. In the mixed functions F5, F6 and F7, although the convergence rate of MCOA is slightly slower than that of COA algorithm in the early stage, it can still find better values than the other eight algorithms except LSHADE in the later stage. For the composite functions F8, F9 and F10, MCOA can find the optimal value faster than the other nine algorithms.

Based on the above, although LSHADE has a stronger ability to find the optimal value in a small number of functions, MCOA can still find the optimal value in most functions in the later stage, and compared with the other eight pair algorithms of the same type, MCOA has more obvious enhancement in optimization ability and avoidance of local optimization, and has better application effect.

4.1.2 Analysis of Wilcoxon rank sum test results

In the comparison experiment, the different effects of multiple algorithms solving the same problem are used to judge whether each algorithm has high efficiency and more obvious influence on solving the current problem, such as the convergence speed of the convergence curve, the fitness value of the optimal solution, the ability to jump out of the local optimum, etc. At present, only the average fitness value, the standard deviation of fitness value and the convergence curve can not be used as the basis for judging whether the performance of the algorithm is efficient. Therefore, the data and images presented by each algorithm in solving the current problem are comprehensively analyzed, and the Wilcoxon rank sum test is used to further verify the difference between MCOA and the other nine comparison algorithms. In this experiment, the significance level is defined as 5%. If its calculated value is less than 5%, it proves that there is a significant difference between the two algorithms, and if it is greater than 5%, it proves that there is no significant difference between the two algorithms. Table 5 shows the Wilcoxon rank-sum test results of the MCOA algorithm and the other nine comparison algorithms. Where the symbols “ + ”, “−” and “ = ” table the performance of MCOA better, worse and equal to the comparison algorithms, respectively.

In the calculation of the function F4 without peak, the value of 1 appears in the comparison of various algorithms such as MCOA, COA, ROA, STOA and other algorithms, indicating that in this function, a variety of algorithms have found the optimal value, there is no obvious difference, which can be ignored. However, in most of the remaining functions, the significance level of MCOA compared with the other nine algorithms is less than 5%, which is a significant difference.

From the overall table, the MCOA algorithm also achieves good results in the Wilcoxon rank-sum test of the IEEE CEC2020 benchmark function, and the contrast ratio with other algorithms is less than 5%, which proves that the MCOA algorithm has a significant difference from the other nine algorithms, and MCOA has better optimization performance. According to the comparison results with the original algorithm, it is proved that MCOA algorithm has a good improvement effect.

4.2 Comparison experiment of single strategy

MCOA adopts two strategies, environment update mechanism and ghost opposition-based learning strategy, to improve COA. In order to prove the effectiveness of these two strategies for algorithm performance optimization, a single strategy comparison experiment is added in this section. In the experiment in this section, EUCOA algorithm which only adds environment update mechanism and GOBLCOA algorithm which only adds ghost opposition-based learning strategy are compared with the basic COA algorithm. The experiments are independently run 30 times in IEEE CEC2020 benchmark test function, and the statistical data obtained are shown in Table  6 . In order to make the table easy to view the statistical results, the poor data in the table will be bolded to make the statistical results more clear and intuitive. It can be seen from the table that among the best fitness values, average fitness values and standard deviation of fitness values of the 10 test functions, GOBLCOA and EUCOA account for less bolded data, while most data of the original algorithm COA are bolded in the table, which effectively proves that both the environment update mechanism and the ghost opposition-based learning strategy play a certain role in COA. The comprehensive performance of COA has been significantly improved.

4.3 Parameter sensitivity analysis of water flow velocity factor c

In order to better prove the influence of flow velocity coefficient on MCOA, we choose different flow velocity coefficient c values for comparison experiments. Table 7 shows the statistical results of 30 independent runs of different water flow velocity coefficients in CEC2020. The bold sections represent the best results. As can be seen from the table, the result obtained by c  = 2 is significantly better than the other values. Only in individual test functions are the results slightly worse. In the F1 function, c  = 5 has the best std. In the F5 function, std is best at c  = 6. Among F10 functions, c  = 5 has the best std. Among the other test functions, both the mean fitness value and std at water flow velocity factor c  = 2 are optimal. Through the above analysis, it is proved that the water flow velocity factor c  = 2 has a good optimization effect.

4.4 Experimental summary

In this section, we first test MCOA's optimization performance on the IEEE CEC2020 benchmark function. The improved MCOA is compared with the original algorithm COA and six other meta-heuristic algorithms in the same environment and the experimental analysis is carried out. Secondly, the rank sum test is used to verify whether there are significant differences between MCOA and the other nine comparison algorithms. Finally, three algorithms, EUCOA with environment update mechanism, GOBLCOA with ghost opposition-based learning strategy, COA and MCOA, are tested to improve performance. These three experimental results show that MCOA has a good ability to find optimal solutions and get rid of local optimal solutions.

5 Constrained engineering design problems

With the new development of the era of big data, the solution process becomes complicated and the calculation results become accurate, and more and more people pay close attention to the dynamic development of the feasibility and practicality of the algorithm, so as to ensure that the algorithm has good practical performance on constrained engineering design problems. In order to verify the optimization effect of MCOA in practical applications, four constrained engineering design problems are selected for application testing of MCOA to evaluate the performance of MCOA in solving practical application problems. Every constrained engineering design problems has a minimization objective function (Papaioannou and Koulocheris 2018 ) that is used to calculate the fitness value for a given problem. In addition, each problem contains a varying number of constraints that are taken into account during the calculation of the objective function. If the constraints are not met, the penalty function (Yeniay 2005 ) is used to adjust the fitness value. However, the processing of constraints is not the focus of our research, our focus is on the optimization of parameters in a convex region composed of constraints (Liu and Lu 2014 ). In order to ensure the fairness of the experiment, the parameters of all experiments in this section are set as follows: the maximum evaluation time MaxFEs is 10,000 and the overall scale N is 30. In each experiment, all the algorithms were analyzed 500 times and the optimal results were obtained.

5.1 Multi-disc clutch braking problem

In the field of vehicle engineering, there is a common constrained engineering design problems multi-disc clutch braking problem, and the purpose of our algorithm is to minimize the mass of the multi-disc clutch by optimizing eight constraints and five variables, so as to improve the performance of the multi-disc clutch. Among them, the five variables are: inner diameter r i , outer diameter r o , brake disc thickness t , driving force F , and surface friction coefficient Z . The specific structure of the multi-disc clutch is shown in Fig.  6 .

Schematic diagram of the multi-disc clutch braking problem

The mathematical formulation of the multi-disc clutch braking problem is as follows.

Objective function:

Subject to:

Variable range:

Other parameters:

After calculation and experiments, the experimental results of the multi-disc clutch braking problem are made into a table as shown in Table  8 . In Table  8 , MCOA concluded that the inner diameter r i  = 70, the outer diameter r 0  = 90, the thickness of the brake disc t  = 1, the driving force F  = 600, and the surface friction coefficient Z  = 2. At this time, the minimum weight obtained is 0.2352424, it is 11.16% higher than the original algorithm. Compared with MCOA, the other five algorithms in the calculation of this problem show that the optimization effect is far lower than that of MCOA.

5.2 Design problem of welding beam

The welded beam design problem is very common in the field of structural engineering and is constrained not only by four decision variables (welding width h , connecting beam length l , beam height t , and connecting beam thickness b ) but also by seven other different conditions. Therefore, it is challenging to solve this problem. The purpose of the optimization algorithm is to achieve the best structural performance of the welded beam and reduce its weight by optimizing the small problems such as the shape, size and layout of the weld under many constraints. The specific structure of the welded beam is shown in Fig.  7 .

Schematic diagram of the welded beam design problem

The mathematical formulation of the welded beam design problem is as follows.

Boundaries:

The experimental results of the welding beam design problem are shown in Table  9 . In the table, the welding width obtained by the MCOA algorithm h  = 0.203034,the length of the connecting beam is l  = 3.310032, the height of the beam is t  = 9.084002, and the thickness of the connecting beam is b  = 0.20578751. At this time, the minimum weight is 1.707524, it is 1.46% higher than the original algorithm. In the welding beam design problem, the weight determines the application effect of the algorithm in the practical problem. The weight of MCOA algorithm is smaller than that of other algorithms. Therefore, the practical application effect of MCOA is much greater than that of other algorithms.

5.3 Design problem of reducer

A reducer is a mechanical device used to reduce the speed of rotation and increase the torque. Among them, gears and bearings are an indispensable part of the reducer design, which have a great impact on the transmission efficiency, running stability and service life of the reducer. The weight of the reducer also determines the use of the reducer. Therefore, we will adjust the number of teeth, shape, radius and other parameters of the gear in the reducer to maximize the role of the reducer, reduce the friction between the parts, and extend the service life of the reducer. In this problem, a total of seven variables are constrained, which are the gear width x 1 , the gear modulus x 2 , the gear teeth x 3 , the length of the first axis between bearings x 4 , the length of the second axis between bearings x 5 , the diameter of the first axis x 6 and the diameter of the second axis x 7 . The specific structure of the reducer is shown in Fig.  8 .

Schematic diagram of the reducer design problem

The mathematical model of the reducer design problem is as follows.

The experimental results of the reducer design problem are shown in Table  10 . From Table  10 , it is known that the gear width calculated by the MCOA algorithm is x 1  = 3.47635, the gear modulus x 2  = 0.7, the gear teeth x 3  = 17, the length of the first axis between the bearings x 4  = 7.3, the length of the second axis between the bearings × 5 = 7.8, and the length of the first axis between the bearings x 5  = 7.8. The diameter of the first axis is x 6  = 3.348620, the diameter of the second axis is x 7  = 5.2768, and the minimum weight is 2988.27135, it is 0.08% higher than the original algorithm. In this experiment, it can be concluded that MCOA has the smallest data among the minimum weights obtained by MCOA and other comparison algorithms in this problem, which proves that MCOA has the best optimization effect in solving such problems.

5.4 Design problem of three-bar truss

Three-bar truss structure is widely used in bridge, building, and mechanical equipment and other fields. However, the size, shape and connection mode of the rod need to be further explored by human beings. Therefore, A 1  =  x 1 and A 2  =  x 2 determined by the pairwise property of the system need to be considered in solving this problem. In addition to this, there will be constraints on the total support load, material cost, and other conditions such as cross-sectional area. The structural diagram of the three-bar truss is shown in Fig.  9 .

Schematic diagram of the three-bar truss design problem

The mathematical formulation of the three-bar truss design problem is as follows.

The experimental results of the three-bar truss design problem are shown in Table  11 , from which it can be concluded that x 1  = 0.7887564and x 2  = 0.4079948of the MCOA algorithm on the three-bar truss design problem. At this time, the minimum weight value is 263.85438633, it is 0.24% higher than the original algorithm. Compared with the minimum weight value of other algorithms, the value of MCOA is the smallest. It is concluded that the MCOA algorithm has a good optimization effect on the three-bar truss design problem.

The experimental results of four constrained engineering design problems show that MCOA has good optimization performance in dealing with problems similar to constrained engineering design problems. In addition, we will also introduce the high-dimensional feature selection problem of the wrapper method, and further judge whether MCOA has good optimization performance and the ability to deal with diversified problems through the classification and processing effect of data.

6 High-dimensional feature selection problem

The objective of feature selection is to eliminate redundant and irrelevant features, thereby obtaining a more accurate model. However, in high-dimensional feature spaces, feature selection encounters challenges such as high computational costs and susceptibility to over-fitting. To tackle these issues, this paper propose novel high-dimensional feature selection methods based on metaheuristic algorithms. These methods aim to enhance the efficiency and effectiveness of feature selection in complex, high-dimensional datasets.

High-dimensional feature selection, as discussed in reference (Ghaemi and Feizi-Derakhshi 2016 ), focuses on processing high-dimensional data to extract relevant features while eliminating redundant and irrelevant ones. This process enhances the model's generalization ability and reduces computational costs. The problem of high-dimensional feature selection is often referred to as sparse modeling, encompassing two primary methods: filter and wrapper. Filter methods, also called classifier-independent methods, can be categorized into univariate and multivariate methods. Univariate methods consider individual features independently, leveraging the correlation and dependence within the data to quickly screen and identify the optimal feature subset. On the other hand, multivariate methods assess relationships between multiple features simultaneously, aiming to comprehensively select the most informative feature combinations. Wrapper methods offer more diverse solutions. This approach treats feature selection as an optimization problem, utilizing specific performance measures of classifiers and objective functions. Wrapper methods continuously explore and evaluate various feature combinations to find the best set of features that maximizes the model’s performance. Unlike filter methods, wrapper methods provide a more customized and problem-specific approach to feature selection.

Filter methods, being relatively single and one-sided, approach the problem of feature selection in a straightforward manner by considering individual features and their relationships within the dataset. However, they might lack the flexibility needed for complex and specific problem scenarios. However, wrapper methods offer tailored and problem-specific solutions. They exhibit strong adaptability, wide applicability, and high relevance to the specific problem at hand. Wrapper methods can be seamlessly integrated into any learning algorithm, allowing for a more customized and targeted approach to feature selection. By treating feature selection as an optimization problem and continuously evaluating different feature combinations, wrapper methods can maximize the effectiveness of the algorithm and optimize its performance to a greater extent compared to filter methods. In summary, wrapper methods provide a more sophisticated and problem-specific approach to feature selection, enabling the algorithm to achieve its maximum potential by selecting the most relevant and informative features for the given task.

6.1 Fitness function

In this subsection, the wrapper method in high-dimensional feature selection is elucidated, employing the classification error rate (CEE) (Wang et al. 2005 ) as an illustrative example. CEE is utilized as the fitness function or objective function to assess the optimization effectiveness of the feature selection algorithm for the problem at hand. Specifically, CEE quantifies the classification error rate when employing the k-nearest-neighbors (KNN) algorithm (Datasets | Feature Selection @ ASU. 2019 ), with the Euclidean distance (ED) (The UCI Machine Learning Repository xxxx) serving as the metric for measuring the distance between the current model being tested and its neighboring models. By using CEE as the fitness function, the wrapper method evaluates different feature subsets based on their performance in the context of the KNN algorithm. This approach enables the algorithm to identify the most relevant features that lead to the lowest classification error rate, thereby optimizing the model's performance. By focusing on the accuracy of classification in a specific algorithmic context, the wrapper method ensures that the selected features are highly tailored to the problem and the chosen learning algorithm. This targeted feature selection process enhances the overall performance and effectiveness of the algorithm in handling high-dimensional data.

where X denotes feat, Y denotes label, both X and Y are specific features in the given data model, and D is the total number of features recorded.

In the experimental setup, each dataset is partitioned into a training set and a test set, with an 80% and 20% ratio. The training set is initially utilized to select the most characteristic features and fine-tune the parameters of the KNN model. Subsequently, the test set is employed to evaluate and calculate the data model and algorithm performance. To address concerns related to fitting ability and overfitting, hierarchical cross-validation with K = 10 was employed in this experiment. In hierarchical cross-validation, the training portion is divided into ten equal-sized subsets. The KNN classifier is trained using 9 out of the 10 folds (K-1 folds) to identify the optimal KNN classifier, while the remaining fold is used for validation purposes. This process is repeated 10 times, ensuring that each subset serves both as a validation set and as part of the training data. This iterative approach is a crucial component of our evaluation methodology, providing a robust assessment of the algorithm's performance. By repeatedly employing replacement validation and folding training, we enhance the reliability and accuracy of our evaluation, enabling a comprehensive analysis of the algorithm's effectiveness across various datasets.

6.2 High-dimensional datasets

In this subsection, the optimization performance of MCOA is assessed using 12 high-dimensional datasets sourced from the Arizona State University (Too et al. 2021 ) and University of California Irvine (UCI) Machine Learning databases (Chandrashekar and Sahin 2014 ). By conducting experiments on these high-dimensional datasets, the results obtained are not only convincing but also pose significant challenges. These datasets authentically capture the intricacies of real-life spatial problems, making the experiments more meaningful and applicable to complex and varied spatial scenarios. For a detailed overview of the 12 high-dimensional datasets, please refer to Table  12 .

6.3 Experimental results and analysis

In order to assess the effectiveness and efficiency of MCOA in feature selection, we conducted comparative tests using MCOA as well as several other algorithms including COA, SSA, PSO, ABC, WSA (Baykasoğlu et al. 2020 ), FPA (Yang 2012 ), and ABO (Qi et al. 2017 ) on 12 datasets. In this section of the experiment, the fitness value of each algorithm was calculated, and the convergence curve, feature selection accuracy (FS Accuracy), and selected feature size for each algorithm were analyzed. Figures 10 , 11 and 12 display the feature selection (FS) convergence curve, FS Accuracy, and selected feature size for the eight algorithms across the 12 datasets. From these figures, it is evident that the optimization ability and prediction accuracy of the MCOA algorithm surpass those of the other seven comparison algorithms. Taking the dataset CLL-SUB-111 as an example in Figs.  11 and 12 , MCOA selected 20 features, while the other seven algorithms selected more than 2000 features. Moreover, the prediction accuracy achieved by MCOA was higher than that of the other seven algorithms. Across all 12 datasets, the comparison figures indicate that the MCOA algorithm consistently outperforms the others. Specifically, the MCOA algorithm tends to select smaller feature subsets, leading to higher prediction accuracy and stronger optimization capabilities. This pattern highlights the superior performance of MCOA in feature selection, demonstrating its effectiveness in optimizing feature subsets for improved prediction accuracy.

Convergence curve of FS

Comparison plot of verification accuracy of eight algorithms

Comparison plots of feature sizes of the eight algorithms

To address the randomness and instability inherent in experiments, a single experiment may not fully demonstrate the effectiveness of algorithm performance. Therefore, we conducted 30 independent experiments using 12 datasets and 8 algorithms. For each algorithm and dataset combination, we calculated the average fitness value, standard deviation of the fitness value, and Friedman rank. Subsequently, the Wilcoxon rank sum test was employed to determine significant differences between the performance of different algorithms across various datasets. Throughout the experiment, a fixed population size of 10 and a maximum of 100 iterations were used. The 12 datasets were utilized to evaluate the 8 algorithms 300 times (tenfold cross-validation × 30 runs). It is essential to note that all algorithms were assessed using the same fitness function derived from the dataset, ensuring a consistent evaluation criterion across the experiments. By conducting multiple independent experiments and statistical analyses, the study aimed to provide a comprehensive and robust assessment of algorithm performance. This approach helps in drawing reliable conclusions regarding the comparative effectiveness of the algorithms under consideration across different datasets, accounting for the inherent variability and randomness in the experimental process.

Table 13 presents the average fitness calculation results from 30 independent experiments for the eight algorithms, it is 55.23% higher than the original algorithm. According to the table, in the Ionosphere dataset, MCOA exhibits the best average fitness, albeit with slightly lower stability compared to ABC. Similarly, in the WarpAR10P dataset, MCOA achieves the best average fitness, with stability slightly lower than COA. After conducting Friedman ranking on the fitness calculation results of the 30 independent experiments, it is concluded that although MCOA shows slightly lower stability in some datasets, it ranks first overall. Among the other seven algorithms, PSO ranks second, ABO ranks third, COA ranks fourth, and ABC, SSA, FPA, and WSA rank fifth to ninth, respectively. These results demonstrate that MCOA exhibits robust optimization performance and high stability in solving high-dimensional feature selection problems. Moreover, MCOA outperforms COA, showcasing its superior improvement in solving these complex problems.

Table 14 presents the accuracy calculation results of the eight algorithms for 30 independent experiments, it is 10.85% higher than the original algorithm. According to the table, the average accuracy of MCOA is the highest across all datasets. Notably, in the Colon dataset, MCOA performs exceptionally well with a perfect average accuracy of 100%. However, in the Ionosphere dataset, MCOA exhibits slightly lower stability compared to ABC, and in the WarpAR10P dataset, it is slightly less stable than COA. Upon conducting Friedman ranking on the average accuracy calculation results of 30 independent experiments, it is evident that MCOA ranks first overall. Among the other seven algorithms, PSO ranks second, ABC ranks third, COA ranks fourth, and ABO, FPA, SSA, and WSA rank fifth to ninth, respectively. These results highlight that MCOA consistently achieves high accuracy and stability in solving high-dimensional feature selection problems. Its superior performance across various datasets underscores its effectiveness and reliability in real-world applications.

Table 15 demonstrates that the MCOA algorithm has shown significant results in the Wilcoxon rank sum test for high-dimensional feature selection fitness. The comparison values with other algorithms are less than 5%, indicating that the MCOA algorithm exhibits significant differences compared to the other seven algorithms. This result serves as evidence that MCOA outperforms the other algorithms, showcasing its superior optimization performance. Additionally, when comparing the results with the original algorithm, it becomes evident that the MCOA algorithm has a substantial and positive impact, demonstrating its effectiveness and improvement over existing methods. These findings underscore the algorithm's potential and its ability to provide substantial enhancements in the field of high-dimensional feature selection.

7 Conclusions and future work

The Crayfish Optimization Algorithm (COA) is grounded in swarm intelligence, drawing inspiration from crayfish behavior to find optimal solutions within a specific range. However, COA’s limitations stem from neglecting crucial survival traits of crayfish, such as crawling against water to discover better aquatic environments. This oversight weakens COA’s search ability, making it susceptible to local optima and hindering its capacity to find optimal solutions. To address these issues, this paper introduces a Modified Crayfish Optimization Algorithm (MCOA). MCOA incorporates an environmental updating mechanism, enabling crayfish to randomly select directions toward better aquatic environments for location updates, enhancing search ability. The addition of the ghost opposition-based learning strategy expands MCOA’s search range and promotes escape from local optima. Experimental validations using IEEE CEC2020 benchmark functions confirm MCOA’s outstanding optimization performance.

Moreover, MCOA’s practical applicability is demonstrated through applications to four constrained engineering problems and high-dimensional feature selection challenges. These experiments underscore MCOA’s efficacy in real-world scenarios, but MCOA can only solve the optimization problem of a single goal. In future studies, efforts will be made to further optimize MCOA and enhance its function. We will exploitation multi-objective version of the algorithm to increase the search ability and convergence of the algorithm through non-dominated sorting, multi-objective selection, crossover and mutation, etc., to solve more complex practical problems. It is extended to wireless sensor network coverage, machine learning, image segmentation and other practical applications.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the support of Fujian Key Lab of Agriculture IOT Application, IOT Application Engineering Research Center of Fujian Province Colleges and Universities, Guiding Science and Technology Projects in Sanming City (2023-G-5), Industry-University Cooperation Project of Fujian Province (2021H6039), Fujian Province Industrial Guidance (Key) Project (2022H0053), Sanming Major Science and Technology Project of Industry-University-Research Collaborative Innovation (2022-G-4), and also the anonymous reviewers and the editor for their careful reviews and constructive suggestions to help us improve the quality of this paper.

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Department of Mechanical Engineering, Bursa Uludağ University, 16059, Görükle, Bursa, Turkey

Ali Riza Yildiz

Department of Computer and Information Science, Linköping University, 58183, Linköping, Sweden

Abdelazim G. Hussien

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Contributions

Heming Jia: Methodology, Formal analysis, Investigation, Resources, Funding acquisition, Project administration; Xuelian Zhou: Investigation, Conceptualization, Software, Data Curation, Writing—Original Draft; Jinrui Zhang: Validation, Conceptualization; Laith Abualigah: Supervision, Writing—Review & Editing; Ali Riza Yildiz: Visualization, Writing—Review & Editing; Abdelazim G. Hussien: Writing—Review & Editing.

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Correspondence to Heming Jia .

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Jia, H., Zhou, X., Zhang, J. et al. Modified crayfish optimization algorithm for solving multiple engineering application problems. Artif Intell Rev 57 , 127 (2024). https://doi.org/10.1007/s10462-024-10738-x

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Accepted : 24 February 2024

Published : 24 April 2024

DOI : https://doi.org/10.1007/s10462-024-10738-x

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