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What Is an Algorithm in Psychology?

Definition, Examples, and Uses

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

algorithm method of problem solving

 James Lacy, MLS, is a fact-checker and researcher.

algorithm method of problem solving

How Does an Algorithm Work?

Examples of algorithms.

  • Reasons to Use Algorithms
  • Potential Pitfalls

Algorithms vs. Heuristics

When solving a problem , choosing the right approach is often the key to arriving at the best solution. In psychology, one of these problem-solving approaches is known as an algorithm. While often thought of purely as a mathematical term, the same type of process can be followed in psychology to find the correct answer when solving a problem or making a decision.

An algorithm is a defined set of step-by-step procedures that provides the correct answer to a particular problem. By following the instructions correctly, you are guaranteed to arrive at the right answer.

At a Glance

Algorithms involve following specific steps in order to reach a solution to a problem. They can be a great tool when you need an accurate solution but tend to be more time-consuming than other methods.

This article discusses how algorithms are used as an approach to problem-solving. It also covers how psychologists compare this approach to other problem-solving methods.

An algorithm is often expressed in the form of a graph, where a square represents each step. Arrows then branch off from each step to point to possible directions that you may take to solve the problem.

In some cases, you must follow a particular set of steps to solve the problem. In other instances, you might be able to follow different paths that will all lead to the same solution.

Algorithms are essential step-by-step approaches to solving a problem. Rather than guessing or using trial-and-error, this approach is more likely to guarantee a specific solution. 

Using an algorithm can help you solve day-to-day problems you face, but it can also help mental health professionals find ways to help people cope with mental health problems.

For example, a therapist might use an algorithm to treat a person experiencing something like anxiety. Because the therapist knows that a particular approach is likely to be effective, they would recommend a series of specific, focused steps as part of their intervention.

There are many different examples of how algorithms can be used in daily life. Some common ones include:

  • A recipe for cooking a particular dish
  • The method a search engine uses to find information on the internet
  • Instructions for how to assemble a bicycle
  • Instructions for how to solve a Rubik's cube
  • A process to determine what type of treatment is most appropriate for certain types of mental health conditions

Doctors and mental health professionals often use algorithms to diagnose mental disorders . For example, they may use a step-by-step approach when they evaluate people.

This might involve asking the individual about their symptoms and their medical history. The doctor may also conduct lab tests, physical exams, or psychological assessments.

Using this information, they then utilize the "Diagnostic and Statistical Manual of Mental Disorders" (DSM-5-TR) to make a diagnosis.

Reasons to Use Algorithms in Psychology

The upside of using an algorithm to solve a problem or make a decision is that yields the best possible answer every time. There are situations where using an algorithm can be the best approach:

When Accuracy Is Crucial

Algorithms can be particularly useful in situations when accuracy is critical. They are also a good choice when similar problems need to be frequently solved.

Computer programs can often be designed to speed up this process. Data then needs to be placed in the system so that the algorithm can be executed for the correct solution.

Artificial intelligence may also be a tool for making clinical assessments in healthcare situations.

When Each Decision Needs to Follow the Same Process

Such step-by-step approaches can be useful in situations where each decision must be made following the same process. Because the process follows a prescribed procedure, you can be sure that you will reach the correct answer each time.

Potential Pitfalls When Using Algorithms

The downside of using an algorithm to solve the problem is that this process tends to be very time-consuming.

So if you face a situation where a decision must be made very quickly, you might be better off using a different problem-solving strategy.

For example, an emergency room doctor making a decision about how to treat a patient could use an algorithm approach. However, this would be very time-consuming and treatment needs to be implemented quickly.

In this instance, the doctor would instead rely on their expertise and past experiences to very quickly choose what they feel is the right treatment approach.

Algorithms can sometimes be very complex and may only apply to specific situations. This can limit their use and make them less generalizable when working with larger populations.

Algorithms can be a great problem-solving choice when the answer needs to be 100% accurate or when each decision needs to follow the same process. A different approach might be needed if speed is the primary concern.

In psychology, algorithms are frequently contrasted with heuristics . Both can be useful when problem-solving, but it is important to understand the differences between them.

What Is a Heuristic?

A heuristic is a mental shortcut that allows people to quickly make judgments and solve problems.

These mental shortcuts are typically informed by our past experiences and allow us to act quickly. However, heuristics are really more of a rule-of-thumb; they don't always guarantee a correct solution.

So how do you determine when to use a heuristic and when to use an algorithm? When problem-solving, deciding which method to use depends on the need for either accuracy or speed.

When to Use an Algorithm

If complete accuracy is required, it is best to use an algorithm. By using an algorithm, accuracy is increased and potential mistakes are minimized.

If you are working in a situation where you absolutely need the correct or best possible answer, your best bet is to use an algorithm. When you are solving problems for your math homework, you don't want to risk your grade on a guess.

By following an algorithm, you can ensure that you will arrive at the correct answer to each problem.

When to Use a Heuristic

On the other hand, if time is an issue, then it may be best to use a heuristic. Mistakes may occur, but this approach allows for speedy decisions when time is of the essence.

Heuristics are more commonly used in everyday situations, such as figuring out the best route to get from point A to point B. While you could use an algorithm to map out every possible route and determine which one would be the fastest, that would be a very time-consuming process. Instead, your best option would be to use a route that you know has worked well in the past.

Psychologists who study problem-solving have described two main processes people utilize to reach conclusions: algorithms and heuristics. Knowing which approach to use is important because these two methods can vary in terms of speed and accuracy.

While each situation is unique, you may want to use an algorithm when being accurate is the primary concern. But if time is of the essence, then an algorithm is likely not the best choice.

Lang JM, Ford JD, Fitzgerald MM. An algorithm for determining use of trauma-focused cognitive-behavioral therapy . Psychotherapy (Chic) . 2010;47(4):554-69. doi:10.1037/a0021184

Stein DJ, Shoptaw SJ, Vigo DV, et al. Psychiatric diagnosis and treatment in the 21st century: paradigm shifts versus incremental integration .  World Psychiatry . 2022;21(3):393-414. doi:10.1002/wps.20998

Bobadilla-Suarez S, Love BC. Fast or frugal, but not both: decision heuristics under time pressure . J Exp Psychol Learn Mem Cogn . 2018;44(1):24-33. doi:10.1037/xlm0000419

Giordano C, Brennan M, Mohamed B, Rashidi P, Modave F, Tighe P. Accessing artificial intelligence for clinical decision-making .  Front Digit Health . 2021;3:645232. doi:10.3389/fdgth.2021.645232

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

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Understanding Algorithms: The Key to Problem-Solving Mastery

algorithm method of problem solving

The world of computer science is a fascinating realm, where intricate concepts and technologies continuously shape the way we interact with machines. Among the vast array of ideas and principles, few are as fundamental and essential as algorithms. These powerful tools serve as the building blocks of computation, enabling computers to solve problems, make decisions, and process vast amounts of data efficiently.

An algorithm can be thought of as a step-by-step procedure or a set of instructions designed to solve a specific problem or accomplish a particular task. It represents a systematic approach to finding solutions and provides a structured way to tackle complex computational challenges. Algorithms are at the heart of various applications, from simple calculations to sophisticated machine learning models and complex data analysis.

Understanding algorithms and their inner workings is crucial for anyone interested in computer science. They serve as the backbone of software development, powering the creation of innovative applications across numerous domains. By comprehending the concept of algorithms, aspiring computer science enthusiasts gain a powerful toolset to approach problem-solving and gain insight into the efficiency and performance of different computational methods.

In this article, we aim to provide a clear and accessible introduction to algorithms, focusing on their importance in problem-solving and exploring common types such as searching, sorting, and recursion. By delving into these topics, readers will gain a solid foundation in algorithmic thinking and discover the underlying principles that drive the functioning of modern computing systems. Whether you’re a beginner in the world of computer science or seeking to deepen your understanding, this article will equip you with the knowledge to navigate the fascinating world of algorithms.

What are Algorithms?

At its core, an algorithm is a systematic, step-by-step procedure or set of rules designed to solve a problem or perform a specific task. It provides clear instructions that, when followed meticulously, lead to the desired outcome.

Consider an algorithm to be akin to a recipe for your favorite dish. When you decide to cook, the recipe is your go-to guide. It lists out the ingredients you need, their exact quantities, and a detailed, step-by-step explanation of the process, from how to prepare the ingredients to how to mix them, and finally, the cooking process. It even provides an order for adding the ingredients and specific times for cooking to ensure the dish turns out perfect.

In the same vein, an algorithm, within the realm of computer science, provides an explicit series of instructions to accomplish a goal. This could be a simple goal like sorting a list of numbers in ascending order, a more complex task such as searching for a specific data point in a massive dataset, or even a highly complicated task like determining the shortest path between two points on a map (think Google Maps). No matter the complexity of the problem at hand, there’s always an algorithm working tirelessly behind the scenes to solve it.

Furthermore, algorithms aren’t limited to specific programming languages. They are universal and can be implemented in any language. This is why understanding the fundamental concept of algorithms can empower you to solve problems across various programming languages.

The Importance of Algorithms

Algorithms are indisputably the backbone of all computational operations. They’re a fundamental part of the digital world that we interact with daily. When you search for something on the web, an algorithm is tirelessly working behind the scenes to sift through millions, possibly billions, of web pages to bring you the most relevant results. When you use a GPS to find the fastest route to a location, an algorithm is computing all possible paths, factoring in variables like traffic and road conditions, to provide you the optimal route.

Consider the world of social media, where algorithms curate personalized feeds based on our previous interactions, or in streaming platforms where they recommend shows and movies based on our viewing habits. Every click, every like, every search, and every interaction is processed by algorithms to serve you a seamless digital experience.

In the realm of computer science and beyond, everything revolves around problem-solving, and algorithms are our most reliable problem-solving tools. They provide a structured approach to problem-solving, breaking down complex problems into manageable steps and ensuring that every eventuality is accounted for.

Moreover, an algorithm’s efficiency is not just a matter of preference but a necessity. Given that computers have finite resources — time, memory, and computational power — the algorithms we use need to be optimized to make the best possible use of these resources. Efficient algorithms are the ones that can perform tasks more quickly, using less memory, and provide solutions to complex problems that might be infeasible with less efficient alternatives.

In the context of massive datasets (the likes of which are common in our data-driven world), the difference between a poorly designed algorithm and an efficient one could be the difference between a solution that takes years to compute and one that takes mere seconds. Therefore, understanding, designing, and implementing efficient algorithms is a critical skill for any computer scientist or software engineer.

Hence, as a computer science beginner, you are starting a journey where algorithms will be your best allies — universal keys capable of unlocking solutions to a myriad of problems, big or small.

Common Types of Algorithms: Searching and Sorting

Two of the most ubiquitous types of algorithms that beginners often encounter are searching and sorting algorithms.

Searching algorithms are designed to retrieve specific information from a data structure, like an array or a database. A simple example is the linear search, which works by checking each element in the array until it finds the one it’s looking for. Although easy to understand, this method isn’t efficient for large datasets, which is where more complex algorithms like binary search come in.

Binary search, on the other hand, is like looking up a word in the dictionary. Instead of checking each word from beginning to end, you open the dictionary in the middle and see if the word you’re looking for should be on the left or right side, thereby reducing the search space by half with each step.

Sorting algorithms, meanwhile, are designed to arrange elements in a particular order. A simple sorting algorithm is bubble sort, which works by repeatedly swapping adjacent elements if they’re in the wrong order. Again, while straightforward, it’s not efficient for larger datasets. More advanced sorting algorithms, such as quicksort or mergesort, have been designed to sort large data collections more efficiently.

Diving Deeper: Graph and Dynamic Programming Algorithms

Building upon our understanding of searching and sorting algorithms, let’s delve into two other families of algorithms often encountered in computer science: graph algorithms and dynamic programming algorithms.

A graph is a mathematical structure that models the relationship between pairs of objects. Graphs consist of vertices (or nodes) and edges (where each edge connects a pair of vertices). Graphs are commonly used to represent real-world systems such as social networks, web pages, biological networks, and more.

Graph algorithms are designed to solve problems centered around these structures. Some common graph algorithms include:

Dynamic programming is a powerful method used in optimization problems, where the main problem is broken down into simpler, overlapping subproblems. The solutions to these subproblems are stored and reused to build up the solution to the main problem, saving computational effort.

Here are two common dynamic programming problems:

Understanding these algorithm families — searching, sorting, graph, and dynamic programming algorithms — not only equips you with powerful tools to solve a variety of complex problems but also serves as a springboard to dive deeper into the rich ocean of algorithms and computer science.

Recursion: A Powerful Technique

While searching and sorting represent specific problem domains, recursion is a broad technique used in a wide range of algorithms. Recursion involves breaking down a problem into smaller, more manageable parts, and a function calling itself to solve these smaller parts.

To visualize recursion, consider the task of calculating factorial of a number. The factorial of a number n (denoted as n! ) is the product of all positive integers less than or equal to n . For instance, the factorial of 5 ( 5! ) is 5 x 4 x 3 x 2 x 1 = 120 . A recursive algorithm for finding factorial of n would involve multiplying n by the factorial of n-1 . The function keeps calling itself with a smaller value of n each time until it reaches a point where n is equal to 1, at which point it starts returning values back up the chain.

Algorithms are truly the heart of computer science, transforming raw data into valuable information and insight. Understanding their functionality and purpose is key to progressing in your computer science journey. As you continue your exploration, remember that each algorithm you encounter, no matter how complex it may seem, is simply a step-by-step procedure to solve a problem.

We’ve just scratched the surface of the fascinating world of algorithms. With time, patience, and practice, you will learn to create your own algorithms and start solving problems with confidence and efficiency.

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How to use algorithms to solve everyday problems

Kara Baskin

May 8, 2017

How can I navigate the grocery store quickly? Why doesn’t anyone like my Facebook status? How can I alphabetize my bookshelves in a hurry? Apple data visualizer and MIT System Design and Management graduate Ali Almossawi solves these common dilemmas and more in his new book, “ Bad Choices: How Algorithms Can Help You Think Smarter and Live Happier ,” a quirky, illustrated guide to algorithmic thinking. 

For the uninitiated: What is an algorithm? And how can algorithms help us to think smarter?

An algorithm is a process with unambiguous steps that has a beginning and an end, and does something useful.

Algorithmic thinking is taking a step back and asking, “If it’s the case that algorithms are so useful in computing to achieve predictability, might they also be useful in everyday life, when it comes to, say, deciding between alternative ways of solving a problem or completing a task?” In all cases, we optimize for efficiency: We care about time or space.

Note the mention of “deciding between.” Computer scientists do that all the time, and I was convinced that the tools they use to evaluate competing algorithms would be of interest to a broad audience.

Why did you write this book, and who can benefit from it?

All the books I came across that tried to introduce computer science involved coding. My approach to making algorithms compelling was focusing on comparisons. I take algorithms and put them in a scene from everyday life, such as matching socks from a pile, putting books on a shelf, remembering things, driving from one point to another, or cutting an onion. These activities can be mapped to one or more fundamental algorithms, which form the basis for the field of computing and have far-reaching applications and uses.

I wrote the book with two audiences in mind. One, anyone, be it a learner or an educator, who is interested in computer science and wants an engaging and lighthearted, but not a dumbed-down, introduction to the field. Two, anyone who is already familiar with the field and wants to experience a way of explaining some of the fundamental concepts in computer science differently than how they’re taught.

I’m going to the grocery store and only have 15 minutes. What do I do?

Do you know what the grocery store looks like ahead of time? If you know what it looks like, it determines your list. How do you prioritize things on your list? Order the items in a way that allows you to avoid walking down the same aisles twice.

For me, the intriguing thing is that the grocery store is a scene from everyday life that I can use as a launch pad to talk about various related topics, like priority queues and graphs and hashing. For instance, what is the most efficient way for a machine to store a prioritized list, and what happens when the equivalent of you scratching an item from a list happens in the machine’s list? How is a store analogous to a graph (an abstraction in computer science and mathematics that defines how things are connected), and how is navigating the aisles in a store analogous to traversing a graph?

Nobody follows me on Instagram. How do I get more followers?

The concept of links and networks, which I cover in Chapter 6, is relevant here. It’s much easier to get to people whom you might be interested in and who might be interested in you if you can start within the ball of links that connects those people, rather than starting at a random spot.

You mention Instagram: There, the hashtag is one way to enter that ball of links. Tag your photos, engage with users who tag their photos with the same hashtags, and you should be on your way to stardom.

What are the secret ingredients of a successful Facebook post?

I’ve posted things on social media that have died a sad death and then posted the same thing at a later date that somehow did great. Again, if we think of it in terms that are relevant to algorithms, we’d say that the challenge with making something go viral is really getting that first spark. And to get that first spark, a person who is connected to the largest number of people who are likely to engage with that post, needs to share it.

With [my first book], “Bad Arguments,” I spent a month pouring close to $5,000 into advertising for that project with moderate results. And then one science journalist with a large audience wrote about it, and the project took off and hasn’t stopped since.

What problems do you wish you could solve via algorithm but can’t?

When we care about efficiency, thinking in terms of algorithms is useful. There are cases when that’s not the quality we want to optimize for — for instance, learning or love. I walk for several miles every day, all throughout the city, as I find it relaxing. I’ve never asked myself, “What’s the most efficient way I can traverse the streets of San Francisco?” It’s not relevant to my objective.

Algorithms are a great way of thinking about efficiency, but the question has to be, “What approach can you optimize for that objective?” That’s what worries me about self-help: Books give you a silver bullet for doing everything “right” but leave out all the nuances that make us different. What works for you might not work for me.

Which companies use algorithms well?

When you read that the overwhelming majority of the shows that users of, say, Netflix, watch are due to Netflix’s recommendation engine, you know they’re doing something right.

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What is Problem Solving Algorithm?, Steps, Representation

  • Post author: Disha Singh
  • Post published: 6 June 2021
  • Post category: Computer Science
  • Post comments: 0 Comments

Table of Contents

  • 1 What is Problem Solving Algorithm?
  • 2 Definition of Problem Solving Algorithm
  • 3.1 Analysing the Problem
  • 3.2 Developing an Algorithm
  • 3.4 Testing and Debugging
  • 4.1 Flowchart
  • 4.2 Pseudo code

What is Problem Solving Algorithm?

Computers are used for solving various day-to-day problems and thus problem solving is an essential skill that a computer science student should know. It is pertinent to mention that computers themselves cannot solve a problem. Precise step-by-step instructions should be given by us to solve the problem.

Problem Solving Algorithm

Thus, the success of a computer in solving a problem depends on how correctly and precisely we define the problem, design a solution (algorithm) and implement the solution (program) using a programming language.

Thus, problem solving is the process of identifying a problem, developing an algorithm for the identified problem and finally implementing the algorithm to develop a computer program.

Definition of Problem Solving Algorithm

These are some simple definition of problem solving algorithm which given below:

Steps for Problem Solving

When problems are straightforward and easy, we can easily find the solution. But a complex problem requires a methodical approach to find the right solution. In other words, we have to apply problem solving techniques.

Problem solving begins with the precise identification of the problem and ends with a complete working solution in terms of a program or software. Key steps required for solving a problem using a computer.

For Example: Suppose while driving, a vehicle starts making a strange noise. We might not know how to solve the problem right away. First, we need to identify from where the noise is coming? In case the problem cannot be solved by us, then we need to take the vehicle to a mechanic.

The mechanic will analyse the problem to identify the source of the noise, make a plan about the work to be done and finally repair the vehicle in order to remove the noise. From the example, it is explicit that, finding the solution to a problem might consist of multiple steps.

Following are Steps for Problem Solving :

Analysing the Problem

Developing an algorithm, testing and debugging.

Steps for Problem Solving

It is important to clearly understand a problem before we begin to find the solution for it. If we are not clear as to what is to be solved, we may end up developing a program which may not solve our purpose.

Thus, we need to read and analyse the problem statement carefully in order to list the principal components of the problem and decide the core functionalities that our solution should have. By analysing a problem, we would be able to figure out what are the inputs that our program should accept and the outputs that it should produce.

It is essential to device a solution before writing a program code for a given problem. The solution is represented in natural language and is called an algorithm. We can imagine an algorithm like a very well-written recipe for a dish, with clearly defined steps that, if followed, one will end up preparing the dish.

We start with a tentative solution plan and keep on refining the algorithm until the algorithm is able to capture all the aspects of the desired solution. For a given problem, more than one algorithm is possible and we have to select the most suitable solution.

After finalising the algorithm, we need to convert the algorithm into the format which can be understood by the computer to generate the desired solution. Different high level programming languages can be used for writing a program. It is equally important to record the details of the coding procedures followed and document the solution. This is helpful when revisiting the programs at a later stage.

The program created should be tested on various parameters. The program should meet the requirements of the user. It must respond within the expected time. It should generate correct output for all possible inputs. In the presence of syntactical errors, no output will be obtained. In case the output generated is incorrect, then the program should be checked for logical errors, if any.

Software industry follows standardised testing methods like unit or component testing, integration testing, system testing, and acceptance testing while developing complex applications. This is to ensure that the software meets all the business and technical requirements and works as expected.

The errors or defects found in the testing phases are debugged or rectified and the program is again tested. This continues till all the errors are removed from the program. Once the software application has been developed, tested and delivered to the user, still problems in terms of functioning can come up and need to be resolved from time to time.

The maintenance of the solution, thus, involves fixing the problems faced by the user, answering the queries of the user and even serving the request for addition or modification of features.

Representation of Algorithms

Using their algorithmic thinking skills, the software designers or programmers analyse the problem and identify the logical steps that need to be followed to reach a solution. Once the steps are identified, the need is to write down these steps along with the required input and desired output.

There are two common methods of representing an algorithm —flowchart and pseudocode. Either of the methods can be used to represent an algorithm while keeping in mind the following:

  • It showcases the logic of the problem solution, excluding any implementational details.
  • It clearly reveals the flow of control during execution of the program.

A flowchart is a visual representation of an algorithm . A flowchart is a diagram made up of boxes, diamonds and other shapes, connected by arrows. Each shape represents a step of the solution process and the arrow represents the order or link among the steps.

A flow chart is a step by step diagrammatic representation of the logic paths to solve a given problem. Or A flowchart is visual or graphical representation of an algorithm .

The flowcharts are pictorial representation of the methods to b used to solve a given problem and help a great deal to analyze the problem and plan its solution in a systematic and orderly manner. A flowchart when translated in to a proper computer language, results in a complete program.

Advantages of Flowcharts:

  • The flowchart shows the logic of a problem displayed in pictorial fashion which felicitates easier checking of an algorithm
  • The Flowchart is good means of communication to other users. It is also a compact means of recording an algorithm solution to a problem.
  • The flowchart allows the problem solver to break the problem into parts. These parts can be connected to make master chart.
  • The flowchart is a permanent record of the solution which can be consulted at a later time.

Differences between Algorithm and Flowchart

Pseudo code.

The Pseudo code is neither an algorithm nor a program. It is an abstract form of a program. It consists of English like statements which perform the specific operations. It is defined for an algorithm. It does not use any graphical representation.

In pseudo code , the program is represented in terms of words and phrases, but the syntax of program is not strictly followed.

Advantages of Pseudocode

  • Before writing codes in a high level language, a pseudocode of a program helps in representing the basic functionality of the intended program.
  • By writing the code first in a human readable language, the programmer safeguards against leaving out any important step. Besides, for non-programmers, actual programs are difficult to read and understand.
  • But pseudocode helps them to review the steps to confirm that the proposed implementation is going to achieve the desire output.

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Problem Solving and Algorithms

Learn a basic process for developing a solution to a problem. Nothing in this chapter is unique to using a computer to solve a problem. This process can be used to solve a wide variety of problems, including ones that have nothing to do with computers.

Problems, Solutions, and Tools

I have a problem! I need to thank Aunt Kay for the birthday present she sent me. I could send a thank you note through the mail. I could call her on the telephone. I could send her an email message. I could drive to her house and thank her in person. In fact, there are many ways I could thank her, but that's not the point. The point is that I must decide how I want to solve the problem, and use the appropriate tool to implement (carry out) my plan. The postal service, the telephone, the internet, and my automobile are tools that I can use, but none of these actually solves my problem. In a similar way, a computer does not solve problems, it's just a tool that I can use to implement my plan for solving the problem.

Knowing that Aunt Kay appreciates creative and unusual things, I have decided to hire a singing messenger to deliver my thanks. In this context, the messenger is a tool, but one that needs instructions from me. I have to tell the messenger where Aunt Kay lives, what time I would like the message to be delivered, and what lyrics I want sung. A computer program is similar to my instructions to the messenger.

The story of Aunt Kay uses a familiar context to set the stage for a useful point of view concerning computers and computer programs. The following list summarizes the key aspects of this point of view.

A computer is a tool that can be used to implement a plan for solving a problem.

A computer program is a set of instructions for a computer. These instructions describe the steps that the computer must follow to implement a plan.

An algorithm is a plan for solving a problem.

A person must design an algorithm.

A person must translate an algorithm into a computer program.

This point of view sets the stage for a process that we will use to develop solutions to Jeroo problems. The basic process is important because it can be used to solve a wide variety of problems, including ones where the solution will be written in some other programming language.

An Algorithm Development Process

Every problem solution starts with a plan. That plan is called an algorithm.

There are many ways to write an algorithm. Some are very informal, some are quite formal and mathematical in nature, and some are quite graphical. The instructions for connecting a DVD player to a television are an algorithm. A mathematical formula such as πR 2 is a special case of an algorithm. The form is not particularly important as long as it provides a good way to describe and check the logic of the plan.

The development of an algorithm (a plan) is a key step in solving a problem. Once we have an algorithm, we can translate it into a computer program in some programming language. Our algorithm development process consists of five major steps.

Step 1: Obtain a description of the problem.

Step 2: analyze the problem., step 3: develop a high-level algorithm., step 4: refine the algorithm by adding more detail., step 5: review the algorithm..

This step is much more difficult than it appears. In the following discussion, the word client refers to someone who wants to find a solution to a problem, and the word developer refers to someone who finds a way to solve the problem. The developer must create an algorithm that will solve the client's problem.

The client is responsible for creating a description of the problem, but this is often the weakest part of the process. It's quite common for a problem description to suffer from one or more of the following types of defects: (1) the description relies on unstated assumptions, (2) the description is ambiguous, (3) the description is incomplete, or (4) the description has internal contradictions. These defects are seldom due to carelessness by the client. Instead, they are due to the fact that natural languages (English, French, Korean, etc.) are rather imprecise. Part of the developer's responsibility is to identify defects in the description of a problem, and to work with the client to remedy those defects.

The purpose of this step is to determine both the starting and ending points for solving the problem. This process is analogous to a mathematician determining what is given and what must be proven. A good problem description makes it easier to perform this step.

When determining the starting point, we should start by seeking answers to the following questions:

What data are available?

Where is that data?

What formulas pertain to the problem?

What rules exist for working with the data?

What relationships exist among the data values?

When determining the ending point, we need to describe the characteristics of a solution. In other words, how will we know when we're done? Asking the following questions often helps to determine the ending point.

What new facts will we have?

What items will have changed?

What changes will have been made to those items?

What things will no longer exist?

An algorithm is a plan for solving a problem, but plans come in several levels of detail. It's usually better to start with a high-level algorithm that includes the major part of a solution, but leaves the details until later. We can use an everyday example to demonstrate a high-level algorithm.

Problem: I need a send a birthday card to my brother, Mark.

Analysis: I don't have a card. I prefer to buy a card rather than make one myself.

High-level algorithm:

Go to a store that sells greeting cards Select a card Purchase a card Mail the card

This algorithm is satisfactory for daily use, but it lacks details that would have to be added were a computer to carry out the solution. These details include answers to questions such as the following.

"Which store will I visit?"

"How will I get there: walk, drive, ride my bicycle, take the bus?"

"What kind of card does Mark like: humorous, sentimental, risqué?"

These kinds of details are considered in the next step of our process.

A high-level algorithm shows the major steps that need to be followed to solve a problem. Now we need to add details to these steps, but how much detail should we add? Unfortunately, the answer to this question depends on the situation. We have to consider who (or what) is going to implement the algorithm and how much that person (or thing) already knows how to do. If someone is going to purchase Mark's birthday card on my behalf, my instructions have to be adapted to whether or not that person is familiar with the stores in the community and how well the purchaser known my brother's taste in greeting cards.

When our goal is to develop algorithms that will lead to computer programs, we need to consider the capabilities of the computer and provide enough detail so that someone else could use our algorithm to write a computer program that follows the steps in our algorithm. As with the birthday card problem, we need to adjust the level of detail to match the ability of the programmer. When in doubt, or when you are learning, it is better to have too much detail than to have too little.

Most of our examples will move from a high-level to a detailed algorithm in a single step, but this is not always reasonable. For larger, more complex problems, it is common to go through this process several times, developing intermediate level algorithms as we go. Each time, we add more detail to the previous algorithm, stopping when we see no benefit to further refinement. This technique of gradually working from a high-level to a detailed algorithm is often called stepwise refinement .

The final step is to review the algorithm. What are we looking for? First, we need to work through the algorithm step by step to determine whether or not it will solve the original problem. Once we are satisfied that the algorithm does provide a solution to the problem, we start to look for other things. The following questions are typical of ones that should be asked whenever we review an algorithm. Asking these questions and seeking their answers is a good way to develop skills that can be applied to the next problem.

Does this algorithm solve a very specific problem or does it solve a more general problem ? If it solves a very specific problem, should it be generalized?

For example, an algorithm that computes the area of a circle having radius 5.2 meters (formula π*5.2 2 ) solves a very specific problem, but an algorithm that computes the area of any circle (formula π*R 2 ) solves a more general problem.

Can this algorithm be simplified ?

One formula for computing the perimeter of a rectangle is:

length + width + length + width

A simpler formula would be:

2.0 * ( length + width )

Is this solution similar to the solution to another problem? How are they alike? How are they different?

For example, consider the following two formulae:

Rectangle area = length * width Triangle area = 0.5 * base * height

Similarities: Each computes an area. Each multiplies two measurements.

Differences: Different measurements are used. The triangle formula contains 0.5.

Hypothesis: Perhaps every area formula involves multiplying two measurements.

Example 4.1: Pick and Plant

This section contains an extended example that demonstrates the algorithm development process. To complete the algorithm, we need to know that every Jeroo can hop forward, turn left and right, pick a flower from its current location, and plant a flower at its current location.

Problem Statement (Step 1)

A Jeroo starts at (0, 0) facing East with no flowers in its pouch. There is a flower at location (3, 0). Write a program that directs the Jeroo to pick the flower and plant it at location (3, 2). After planting the flower, the Jeroo should hop one space East and stop. There are no other nets, flowers, or Jeroos on the island.

Analysis of the Problem (Step 2)

The flower is exactly three spaces ahead of the jeroo.

The flower is to be planted exactly two spaces South of its current location.

The Jeroo is to finish facing East one space East of the planted flower.

There are no nets to worry about.

High-level Algorithm (Step 3)

Let's name the Jeroo Bobby. Bobby should do the following:

Get the flower Put the flower Hop East

Detailed Algorithm (Step 4)

Get the flower Hop 3 times Pick the flower Put the flower Turn right Hop 2 times Plant a flower Hop East Turn left Hop once

Review the Algorithm (Step 5)

The high-level algorithm partitioned the problem into three rather easy subproblems. This seems like a good technique.

This algorithm solves a very specific problem because the Jeroo and the flower are in very specific locations.

This algorithm is actually a solution to a slightly more general problem in which the Jeroo starts anywhere, and the flower is 3 spaces directly ahead of the Jeroo.

Java Code for "Pick and Plant"

A good programmer doesn't write a program all at once. Instead, the programmer will write and test the program in a series of builds. Each build adds to the previous one. The high-level algorithm will guide us in this process.

FIRST BUILD

To see this solution in action, create a new Greenfoot4Sofia scenario and use the Edit Palettes Jeroo menu command to make the Jeroo classes visible. Right-click on the Island class and create a new subclass with the name of your choice. This subclass will hold your new code.

The recommended first build contains three things:

The main method (here myProgram() in your island subclass).

Declaration and instantiation of every Jeroo that will be used.

The high-level algorithm in the form of comments.

The instantiation at the beginning of myProgram() places bobby at (0, 0), facing East, with no flowers.

Once the first build is working correctly, we can proceed to the others. In this case, each build will correspond to one step in the high-level algorithm. It may seem like a lot of work to use four builds for such a simple program, but doing so helps establish habits that will become invaluable as the programs become more complex.

SECOND BUILD

This build adds the logic to "get the flower", which in the detailed algorithm (step 4 above) consists of hopping 3 times and then picking the flower. The new code is indicated by comments that wouldn't appear in the original (they are just here to call attention to the additions). The blank lines help show the organization of the logic.

By taking a moment to run the work so far, you can confirm whether or not this step in the planned algorithm works as expected.

THIRD BUILD

This build adds the logic to "put the flower". New code is indicated by the comments that are provided here to mark the additions.

FOURTH BUILD (final)

Example 4.2: replace net with flower.

This section contains a second example that demonstrates the algorithm development process.

There are two Jeroos. One Jeroo starts at (0, 0) facing North with one flower in its pouch. The second starts at (0, 2) facing East with one flower in its pouch. There is a net at location (3, 2). Write a program that directs the first Jeroo to give its flower to the second one. After receiving the flower, the second Jeroo must disable the net, and plant a flower in its place. After planting the flower, the Jeroo must turn and face South. There are no other nets, flowers, or Jeroos on the island.

Jeroo_2 is exactly two spaces behind Jeroo_1.

The only net is exactly three spaces ahead of Jeroo_2.

Each Jeroo has exactly one flower.

Jeroo_2 will have two flowers after receiving one from Jeroo_1. One flower must be used to disable the net. The other flower must be planted at the location of the net, i.e. (3, 2).

Jeroo_1 will finish at (0, 1) facing South.

Jeroo_2 is to finish at (3, 2) facing South.

Each Jeroo will finish with 0 flowers in its pouch. One flower was used to disable the net, and the other was planted.

Let's name the first Jeroo Ann and the second one Andy.

Ann should do the following: Find Andy (but don't collide with him) Give a flower to Andy (he will be straight ahead) After receiving the flower, Andy should do the following: Find the net (but don't hop onto it) Disable the net Plant a flower at the location of the net Face South
Ann should do the following: Find Andy Turn around (either left or right twice) Hop (to location (0, 1)) Give a flower to Andy Give ahead Now Andy should do the following: Find the net Hop twice (to location (2, 2)) Disable the net Toss Plant a flower at the location of the net Hop (to location (3, 2)) Plant a flower Face South Turn right

The high-level algorithm helps manage the details.

This algorithm solves a very specific problem, but the specific locations are not important. The only thing that is important is the starting location of the Jeroos relative to one another and the location of the net relative to the second Jeroo's location and direction.

Java Code for "Replace Net with Flower"

As before, the code should be written incrementally as a series of builds. Four builds will be suitable for this problem. As usual, the first build will contain the main method, the declaration and instantiation of the Jeroo objects, and the high-level algorithm in the form of comments. The second build will have Ann give her flower to Andy. The third build will have Andy locate and disable the net. In the final build, Andy will place the flower and turn East.

This build creates the main method, instantiates the Jeroos, and outlines the high-level algorithm. In this example, the main method would be myProgram() contained within a subclass of Island .

This build adds the logic for Ann to locate Andy and give him a flower.

This build adds the logic for Andy to locate and disable the net.

This build adds the logic for Andy to place a flower at (3, 2) and turn South.

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Engineering LibreTexts

1: Algorithmic Problem Solving

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Unit Objectives

Upon completion of this unit the learner should be able to:

  • describe an algorithm
  • explain the relationship between data and algorithm
  • outline the characteristics of algorithms
  • apply pseudo codes and flowcharts to represent algorithms

Unit Introduction

This unit introduces learners to data structures and algorithm course. The unit is on the different data structures and their algorithms that can help implement the different data structures in the computer. The application of the different data structures is presented by using examples of algorithms and which are not confined to a particular computer programming language.

  • Data: the structural representation of logical relationships between elements of data
  • Algorithm: finite sequence of steps for accomplishing some computational task
  • Pseudo code: an informal high-level description of the operating principle of a computer program or other algorithm
  • Flow chart: diagrammatic representation illustrates a solution model to a given problem.

Learning Activities

  • 1.1: Activity 1 - Introduction to Algorithms and Problem Solving In this learning activity section, the learner will be introduced to algorithms and how to write algorithms to solve tasks faced by learners or everyday problems. Examples of the algorithm are also provided with a specific application to everyday problems that the learner is familiar with. The learners will particularly learn what is an algorithm, the process of developing a solution for a given task, and finally examples of application of the algorithms are given.
  • 1.2: Activity 2 - The characteristics of an algorithm This section introduces the learners to the characteristics of algorithms. These characteristics make the learner become aware of what to ensure is basic, present and mandatory for any algorithm to qualify to be one. It also exposes the learner to what to expect from an algorithm to achieve or indicate. Key expectations are: the fact that an algorithm must be exact, terminate, effective, general among others.
  • 1.3: Activity 3 - Using pseudo-codes and flowcharts to represent algorithms The student will learn how to design an algorithm using either a pseudo code or flowchart. Pseudo code is a mixture of English like statements, some mathematical notations and selected keywords from a programming language. It is one of the tools used to design and develop the solution to a task or problem. Pseudo codes have different ways of representing the same thing and emphasis is on the clarity and not style.
  • 1.4: Unit Summary In this unit, you have seen what an algorithm is. Based on this knowledge, you should now be able to characterize an algorithm by stating its properties. We have explored the different ways of representing an algorithm such as using human language, pseudo codes and flow chart. You should now be able to present solutions to problems in form of an algorithm.

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  • What Is an Algorithm? | Definition & Examples

What Is an Algorithm? | Definition & Examples

Published on August 9, 2023 by Kassiani Nikolopoulou . Revised on August 29, 2023.

An algorithm is a set of steps for accomplishing a task or solving a problem. Typically, algorithms are executed by computers, but we also rely on algorithms in our daily lives. Each time we follow a particular step-by-step process, like making coffee in the morning or tying our shoelaces, we are in fact following an algorithm.

In the context of computer science , an algorithm is a mathematical process for solving a problem using a finite number of steps. Algorithms are a key component of any computer program and are the driving force behind various systems and applications, such as navigation systems, search engines, and music streaming services.

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Table of contents

What is an algorithm, how do algorithms work, examples of algorithms, other interesting articles, frequently asked questions about algorithms.

An algorithm is a sequence of instructions that a computer must perform to solve a well-defined problem. It essentially defines what the computer needs to do and how to do it. Algorithms can instruct a computer how to perform a calculation, process data, or make a decision.

The best way to understand an algorithm is to think of it as a recipe that guides you through a series of well-defined actions to achieve a specific goal. Just like a recipe produces a replicable result, algorithms ensure consistent and reliable outcomes for a wide range of tasks in the digital realm.

And just like there are numerous ways to make, for example, chocolate chip cookies by following different steps or using slightly different ingredients, different algorithms can be designed to solve the same problem, with each taking a distinct approach but achieving the same result.

Algorithms are virtually everywhere around us. Examples include the following:

  • Search engines rely on algorithms to find and present relevant results as quickly as possible
  • Social media platforms use algorithms to prioritize the content that we see in our feeds, taking into account factors like our past behavior, the popularity of posts, and relevance.
  • With the help of algorithms, navigation apps determine the most efficient route for us to reach our destination.
  • It must be correct . In other words, it should take a given problem and provide the right answer or result, even if it stops working due to an error.
  • It must consist of clear, practical steps that can be completed in a limited time, whether by a person or the machine that must execute the algorithm. For example, the instructions in a cookie recipe might be considered sufficiently concrete for a human cook, but they would not be specific enough for programming an automated cookie-making machine.
  • There should be no confusion about which step comes next , even if choices must be made (e.g., when using “if” statements).
  • It must have a set number of steps (not an infinite number) that can be managed using loops (statements describing repeated actions or iterations).
  • It must eventually reach an endpoint and not get stuck in a never-ending loop.

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algorithm method of problem solving

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Algorithms use a set of initial data or input , process it through a series of logical steps or rules, and produce the output (i.e., the outcome, decision, or result).

Algorithm boxes

If you want to make chocolate chip cookies, for instance, the input would be the ingredients and quantities, the process would be the recipe you choose to follow, and the output would be the cookies.

Algorithms are eventually expressed in a programming language that a computer can process. However, when an algorithm is being created, it will be people, not a computer, who will need to understand it. For this reason, as a first step, algorithms are written as plain instructions.

  • Input: the input data is a single-digit number (e.g., 5).
  • Transformation/processing: the algorithm takes the input (number 5) and performs the specific operation (i.e., multiplies the number by itself).
  • Output: the result of the calculation is the square of the input number, which, in this case, would be 25 (since 5 * 5 = 25).

We could express this as an algorithm in the following way:

Algorithm: Calculate the square of a number

  • Input the number (N) whose square you want to find.
  • Multiply the number (N) by itself.
  • Store the result of the multiplication in a variable (result).
  • Output the value of the variable (result), which represents the square of the input number.

It is important to keep in mind that an algorithm is not the same as a program or code. It is the logic or plan for solving a problem represented as a simple step-by-step description. Code is the implementation of the algorithm in a specific programming language (like C++ or Python), while a program is an implementation of code that instructs a computer on how to execute an algorithm and perform a task.

Instead of telling a computer exactly what to do, some algorithms allow computers to learn on their own and improve their performance on a specific task. These machine learning algorithms use data to identify patterns and make predictions or conduct data mining to uncover hidden insights in data that can inform business decisions.

Broadly speaking, there are three different types of algorithms:

  • Linear sequence algorithms follow a specific set or steps, one after the other. Just like following a recipe, each step depends on the success of the previous one.
  • For example, in the context of a cookie recipe, you would include the step “if the dough is too sticky, you might need to refrigerate it.”
  • For example, a looping algorithm could be used to handle the process of making multiple cookies from a single batch of dough. The algorithm would repeat a specific set of instructions to form and bake cookies until all the dough has been used.

Algorithms are fundamental tools for problem-solving in both the digital world and many real-life scenarios. Each time we try to solve a problem by breaking it down into smaller, manageable steps, we are in fact using algorithmic thinking.

  • Identify which clothes are clean.
  • Consider the weather forecast for the day.
  • Consider the occasion for which you are getting dressed (e.g., work or school etc.).
  • Consider personal preferences (e.g., style or which items match).

In mathematics, algorithms are standard methods for performing calculations or solving equations because they are efficient, reliable, and applicable to various situations.

Suppose you want to add the numbers 345 and 278. You would follow a set of steps (i.e., the standard algorithm for addition):

  • Write down the numbers so the digits align.
  • Start from the rightmost digits (the ones place) and add them together: 5 + 8 = 13. Write down the 3 and carry over the 1 to the next column.
  • Move to the next column (the tens place) and add the digits along with the carried-over value: 4 + 7 + 1 = 12. Write down the 2 and carry over the 1 to the next column.
  • Move to the leftmost column (the hundreds place) and add the digits along with the carried-over value: 3 + 2 + 1 = 6. Write down the 6.

The final result is 623

Algorithm calculation example

Navigation systems are another example of the use of algorithms. Such systems use algorithms to help you find the easiest and fastest route to your destination while avoiding traffic jams and roadblocks.

If you want to know more about ChatGPT, AI tools , fallacies , and research bias , make sure to check out some of our other articles with explanations and examples.

  • ChatGPT vs human editor
  • ChatGPT citations
  • Is ChatGPT trustworthy?
  • Using ChatGPT for your studies
  • Sunk cost fallacy
  • Straw man fallacy
  • Slippery slope fallacy
  • Red herring fallacy
  • Ecological fallacy
  • Logical fallacy

Research bias

  • Implicit bias
  • Framing bias
  • Cognitive bias
  • Optimism bias
  • Hawthorne effect
  • Unconscious bias

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In computer science, an algorithm is a list of unambiguous instructions that specify successive steps to solve a problem or perform a task. Algorithms help computers execute tasks like playing games or sorting a list of numbers. In other words, computers use algorithms to understand what to do and give you the result you need.

Algorithms and artificial intelligence (AI) are not the same, however they are closely related.

  • Artificial intelligence is a broad term describing computer systems performing tasks usually associated with human intelligence like decision-making, pattern recognition, or learning from experience.
  • Algorithms are the instructions that AI uses to carry out these tasks, therefore we could say that algorithms are the building blocks of AI—even though AI involves more advanced capabilities beyond just following instructions.

Algorithms and computer programs are sometimes used interchangeably, but they refer to two distinct but interrelated concepts.

  • An algorithm is a step-by-step instruction for solving a problem that is precise yet general.
  • Computer programs are specific implementations of an algorithm in a specific programming language. In other words, the algorithm is the high-level description of an idea, while the program is the actual implementation of that idea.

Algorithms are valuable to us because they:

  • Form the basis of much of the technology we use in our daily lives, from mobile apps to search engines.
  • Power innovations in various industries that augment our abilities (e.g., AI assistants or medical diagnosis).
  • Help analyze large volumes of data, discover patterns and make informed decisions in a fast and efficient way, at a scale humans are simply not able to do.
  • Automate processes. By streamlining tasks, algorithms increase efficiency, reduce errors, and save valuable time.

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Learning Algorithms by George Heineman

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Chapter 1. Problem Solving

What is an algorithm.

Explaining how an algorithm works is like telling a story. Each algorithm introduces a novel concept or innovation that improves upon ordinary solutions. In this chapter I explore several solutions to a simple problem to explain the factors that affect an algorithm’s performance. Along the way I introduce techniques used to analyze an algorithm’s performance independent of its implementation , though I will always provide empirical evidence from actual implementations.

An algorithm is a step-by-step problem-solving method implemented as a computer program that returns a correct result in a predictable amount of time. The study of algorithms is concerned with both correctness (will this algorithm work for all input?) and performance (is this the most efficient way to solve this problem?).

Let’s walk through an example of a problem-solving method to see what this looks like in practice. What if you wanted to find the largest value in an unordered list? Each Python list in Figure 1-1 is a problem instance , that is, the input processed by an algorithm (shown as a cylinder); the correct answer appears on the right. How is this algorithm implemented? How would it perform on different problem instances? Can you predict the time needed to find the largest value in a list of one million values?

Algorithm Processing

Figure 1-1. Three different problem instances processed by an algorithm

An algorithm is more than just a problem-solving method; the program also needs to complete in a predictable amount of time. The built-in Python function max() already solves this problem. Now, it can be hard to predict an algorithm’s performance on problem instances containing random data, so it’s worth identifying problem instances that are carefully constructed.

Table 1-1 shows the results of timing max() on two kinds of problem instances of size N: one where the list contains ascending integers and one where the list contains descending integers. While your execution may yield different results in the table, based on the configuration of your computing system, you can verify the following two statements:

The timing for max() on ascending values is always slower than on descending values once N is large enough .

As N increases ten-fold in subsequent rows, the corresponding time for max() also appears to increase ten-fold, with some deviation, as is to be expected from live performance trials.

For this problem, the maximum value is returned, and the input is unchanged. In some cases, the algorithm updates the problem instance directly instead of computing a new value—for example, sorting a list of values, as you will see in Chapter 5 . In this book, N represents the size of a problem instance.

When it comes to timing:

You can’t predict in advance the value of T (100,000)—that is, the time required by the algorithm to solve a problem instance of size 100,000—because computing platforms vary, and different programming languages may be used.

However, once you empirically determine T (10,000), you can predict T (100,000)—that is, the time to solve a problem instance ten times larger—though the prediction will inevitably be inaccurate to an extent.

When designing an algorithm, the primary challenge is to ensure it is correct and works for all input . I will spend more time in Chapter 2 explaining how to analyze and compare the behavior of different algorithms that solve the exact same problem. The field of algorithm analysis is tied to the study of interesting, relevant problems that arise in real life. While the mathematics of algorithms can be challenging to understand, I will provide specific examples to always connect the abstract concepts with real-world problems.

The standard way to judge the efficiency of an algorithm is to count how many computing operations it requires. But this is exceptionally hard to do! Computers have a central processing unit (CPU) that executes machine instructions that perform mathematical computations (like add and multiply), assign values to CPU registers, and compare two values with each other. Modern programming languages (like C or C++) are compiled into machine instructions. Other languages (like Python or Java) are compiled into an intermediate byte code representation. The Python interpreter (which is itself a C program) executes the byte code, while built-in functions, such as min() and max() , are implemented in C and ultimately compiled into machine instructions for execution.

It is nearly impossible to count the total number of executed machine instructions for an algorithm, not to mention that modern day CPUs can execute billions of instructions per second! Instead, I will count the number of times a key operation is invoked for each algorithm, which could be “the number of times two values in an array are compared with each other” or “how many times a function is called.” In this discussion of max() , the key operation is “how many times the less-than (<) operator is invoked.” I will expand on this counting principle in Chapter 2 .

Now is a good time to lift up the hood on the max() algorithm to see why it behaves the way it does.

Finding the Largest Value in an Arbitrary List

Consider the flawed Python implementation in Listing 1-1 that attempts to find the largest value in an arbitrary list containing at least one value by comparing each value in A against my_max , updating my_max as needed when larger values are found.

Listing 1-1. Flawed implementation to locate largest value in list

1

my_max is a variable that holds the maximum value; here my_max is initialized to 0.

The for loop defines a variable v that iterates over each element in A . The if statement executes once for each value, v .

Update my_max if v is larger.

Central to this solution is the less-than operator (<) that compares two numbers to determine whether a value is smaller than another. In Figure 1-2 , as v takes on successive values from A , you can see that my_max is updated three times to determine the largest value in A . flawed() determines the largest value in A , invoking less-than six times, once for each of its values. On a problem instance of size N, flawed() invokes less-than N times.

LargestVisualization

Figure 1-2. Visualizing the execution of flawed()

This implementation is flawed because it assumes that at least one value in A is greater than 0. Computing flawed([–5,–3,–11]) returns 0, which is incorrect. One common fix is to initialize my_max to the smallest possible value, such as my_max = float('-inf') . This approach is still flawed since it would return this value if A is the empty list [] . Let’s fix this defect now.

The Python statement range(x,y) produces the integers from x up to, but not including, y . You can also request range(x,y,–1) , which produces the integers from x counting down to, but not including, y . Thus list(range(1,7)) produces [1,2,3,4,5,6] , and list(range(5,0,–1)) produces [5,4,3,2,1] . You can count by arbitrary increments, thus list(range(1,10,2)) produces [1,3,5,7,9] using a difference of 2 between values.

Counting Key Operations

Since the largest value must actually be contained in A , the correct largest() function in Listing 1-2 selects the first value of A as my_max , checking other values to see if any value is larger.

Listing 1-2. Correct function to find largest value in list

Set my_max to the first value in A , found at index position 0.

idx takes on integer values from 1 up to, but not including, len ( A ).

Update my_max if the value in A at position idx is larger.

If you invoke largest() or max() with an empty list, it will raise a ValueError: list index out of range exception. These runtime exceptions are programmer errors, reflecting a failure to understand that largest() requires a list with at least one value.

Now that we have a correct Python implementation of our algorithm, can you determine how many times less-than is invoked in this new algorithm? Right! N – 1 times. We have fixed the flaw in the algorithm and improved its performance (admittedly, by just a tiny bit).

Why is it important to count the uses of less-than? This is the key operation used when comparing two values. All other program statements (such as for or while loops) are arbitrary choices during implementation, based on the program language used. We will expand on this idea in the next chapter, but for now counting key operations is sufficient.

Models Can Predict Algorithm Performance

What if someone shows you a different algorithm for this same problem? How would you determine which one to use? Consider the alternate() algorithm in Listing 1-3 that repeatedly checks each value in A to see if it is larger than or equal to all other values in the same list. Will this algorithm return the correct result? How many times does it invoke less-than on a problem of size N?

Listing 1-3. A different approach to locating largest value in A

4

When iterating over A , assume each value, v , could be the largest.

If v is smaller than another value, x , stop and record that v is not greatest.

If v_is_largest is true , return v since it is the maximum value in A .

If A is an empty list, return None .

alternate() attempts to find a value, v , in A such that no other value, x , in A is greater. The implementation uses two nested for loops. This time it’s not so simple to compute how many times less-than is invoked, because the inner for loop over x stops as soon as an x is found that is greater than v . Also, the outer for loop over v stops once the maximum value is found. Figure 1-3 visualizes executing alternate() on our list example.

Alternate Visualization

Figure 1-3. Visualizing the execution of alternate()

For this problem instance, less-than is invoked 14 times. But you can see that this total count depends on the specific values in the list A . What if the values were in a different order? Can you think of an arrangement of values that requires the least number of less-than invocations? Such a problem instance would be considered a best case for alternate() . For example, if the first value in A is the largest of all N values, then the total number of calls to less-than is always N. To summarize:

A problem instance of size N that requires the least amount of work performed by an algorithm

A problem instance of size N that demands the most amount of work

Let’s try to identify a worst case problem instance for alternate() that requires the most number of calls to less-than. More than just ensuring that the largest value is the last value in A , in a worst case problem instance for alternate() , the values in A must appear in ascending order.

Figure 1-4 visualizes a best case on the top where p = [9,5,2,1,3,4] and a worst case on the bottom where p = [1,2,3,4,5,9] .

Best and Worst Case

Figure 1-4. Visualizing the execution of alternate() on best and worst cases

In the best case , there are six calls to less-than; if there were N values in a best case , then the total number of invocations to less-than would be N. It’s a bit more complicated for the worst case . In Figure 1-4 you can see there are a total of 26 calls to less-than when the list of N values is in ascending sorted order. With a little bit of mathematics, I can show that for N values, this count will always be (N 2 + 3N – 2)/2.

Table 1-2 presents empirical evidence on largest() and alternate() on worst case problem instances of size N.

For small problem instances, it doesn’t seem bad, but as the problem instances double in size, the number of less-than invocations for alternate() essentially quadruples, far surpassing the count for largest() . The next two columns in Table 1-2 show the performance of these two implementations on 100 random trials of problem instances of size N. The completion time for alternate() quadruples as well.

I measure the time required by an algorithm to process random problem instances of size N. From this set of runs, I select the quickest completion time (i.e., the smallest). This is preferable to simply averaging the total running time over all runs, which might skew the results.

Throughout this book, I am going to present tables, like Table 1-2 , containing the total number of executions of a key operation (here, the less-than operator) as well as the runtime performance. Each row will represent a different problem instance size, N. Read the table from top to bottom to see how the values in each column change as the problem instance size doubles.

Counting the number of less-than invocations explains the behaviors of largest() and alternate() . As N doubles, the number of calls to less-than doubles for largest() but quadruples for alternate() . This behavior is consistent and you can use this information to predict the runtime performance of these two algorithms on larger-sized problems. Figure 1-5 plots the count of less-than invocations by alternate() (using the y-axis on the left) against its runtime performance (using the y-axis on the right), showing how directly they correlate with each other.

Tournament

Figure 1-5. Relationship between the number of less-than operations and runtime performance

Congratulations! You’ve just performed a key step in algorithmic analysis: judging the relative performance of two algorithms by counting the number of times a key operation is performed. You can certainly go and implement both variations (as I did) and measure their respective runtime performance on problem instances that double in size; but it actually isn’t necessary since the model predicts this behavior and confirms that largest() is the more efficient algorithm of the two.

largest() and max() are implementations of the same algorithm, but as Table 1-3 shows, largest() is always significantly slower than max() , typically four times slower. The reason is that Python is an interpreted language, which means it is compiled to an intermediate byte code that is executed by a Python interpreter. Built-in functions, such as max() , will always outperform Python code written for the same purpose because the built-in function is implemented within the interpreter. What you should observe is that in all cases, as N doubles, the corresponding performance of largest() and max() —for both best case and worst case —also doubles.

Table 1-3 shows it is possible to predict the time required to solve problem instances of increasing size. Once you know the runtime performance of largest() or max() on a best or worst case problem instance of size N, you can predict that the runtime performance will double as N doubles. Now let’s change the problem slightly to make it more interesting.

Find Two Largest Values in an Arbitrary List

Devise an algorithm that finds the two largest values in an arbitrary list. Perhaps you can modify the existing largest() algorithm with just a few tweaks. Why don’t you take a stab at solving this modified problem and come back here with your solution? Listing 1-4 contains my solution.

Listing 1-4. Find two largest values by tweaking largest() approach

Ensure my_max and second are the first two values from A in descending order.

If A[idx] is a newly found maximum value, then set my_max to A[idx] , and second becomes the old my_max .

If A[idx] is larger than second (but smaller than my_max ), only update second .

largest_two() feels similar to largest() : compute my_max and second to be the first two values in A , properly ordered. Then for each of the remaining values in A (how many? N – 2, right!), if you find an A[idx] larger than my_max , adjust both my_max and second , otherwise check to see whether only second needs updating.

Counting the number of times less-than is invoked is more complicated because, once again, it depends on the values in the problem instance .

largest_two() performs the fewest less-than invocations when the condition of the if statement inside the for loop is true. When A contains values in ascending order, this less-than is always true, so it is invoked N – 2 times; don’t forget to add 1 because of the use of less-than at the beginning of the function. In the best case , therefore, you only need N – 1 invocations of less-than to determine the top two values. The less-than in the elif condition is never used in the best case .

For largest_two() , can you construct a worst case problem instance? Try it yourself: it happens whenever the less-than in the if condition within the for loop is False .

I bet you can see that whenever A contains values in descending order, largest_two() requires the most invocations of less-than. In particular, for the worst case , less-than is used twice each time through the for loop, or 1 + 2 × (N – 2) = 2N – 3 times. Somehow this feels right, doesn’t it? If you need to use less-than N – 1 times to find the largest value in A , perhaps you truly do need an additional N – 2 less-than invocations (leaving out the largest value, of course) to also find the second-largest value.

To summarize the behavior of largest_two() :

For best case , it finds both values with N – 1 less-than invocations.

For worst case , it finds both values with 2N – 3 less-than invocations.

Are we done? Is this the “best” algorithm to solve the problem of finding the two largest values in an arbitrary list? We can choose to prefer one algorithm over another based on a number of factors:

Does an algorithm need to make a copy of the original problem instance?

How few lines of code must the programmer write?

Does the algorithm modify the input provided by the problem instance in place , or does it leave it alone?

Does an algorithm outperform the competition, independent of the provided input?

Let’s investigate three different algorithms to solve this exact same problem, shown in Listing 1-5 . sorting_two() creates a new list containing the values in A in descending order, grabs the first two values, and returns them as a tuple. double_two() uses max() to find the maximum value in A , removes it from a copy of A , and then uses max() of that reduced list to find the second largest. mutable_two() finds the location of the largest value in A and removes it from A ; then it sets second to the largest value remaining in A before reinserting the my_max value into its original location. The first two algorithms require extra storage, while the third modifies its input: all three only work on problem instances containing more than one value.

Listing 1-5. Three different approaches using Python utilities

5

Create a new list by sorting A in descending order and return its top two values.

Use built-in max () function to find largest.

Create a copy of the original A , and remove my_max .

Return a tuple containing my_max and the largest value in copy .

This Python trick finds the index location of the maximum value in A , rather than the value itself.

Record my_max value and delete it from A .

Now find max() of remaining values.

Restore A by inserting my_max to its original location.

These different approaches do not directly use less-than because they rely on existing Python libraries. Both sorting_two() and double_two() make a copy of the array, A , which seems unnecessary, since largest_two() doesn’t do this. In addition, it seems excessive to sort the entire list just to find the top two largest values. In the same way that I count operations when analyzing runtime performance, I will evaluate the extra storage used by an algorithm—for both of these approaches, the amount of storage is directly proportional to N. The third approach, mutable_two() , briefly updates A by deleting its maximum value, only to add it back later. The caller might be surprised to see that the original list was modified.

With a bit of Python expertise, I can compute exactly how many times less-than is invoked using a special RecordedItem class. 1 Table 1-4 shows that double_two() invokes the most less-than operations when the values are in ascending order, while largest_two() (and others) perform the most less-than operations when the values are in descending order. In the last column, labeled “Alternating,” the 524,288 values are arranged with even numbers in ascending order, alternating with odd numbers in descending order: for N = 8, the input would be [0,7,2,5,4,3,6,1] .

The tournament_two() algorithm I describe next has the fewest number of less-than invocations regardless of input. Basketball fans will find its logic familiar.

If you determine the worst case problem instance for an algorithm that solves a given problem, perhaps a different algorithm solving the same problem would not be so negatively affected by that problem instance. Different algorithms can have different weaknesses that you can uncover with diligent analysis.

Tournament Algorithm

A single-elimination tournament consists of a number of teams competing to be the champion. Ideally, the number of teams is a power of 2, like 16 or 64. The tournament is built from a sequence of rounds where all remaining teams in the tournament are paired up to play a match; match losers are eliminated, while winners advance to the next round. The final team remaining is the tournament champion.

Consider the problem instance p = [3,1,4,1,5,9,2,6] with N = 8 values. Figure 1-6 shows the single-elimination tournament whose initial round compares eight neighboring values using less-than; larger values advance in the tournament. 2 In the Elite Eight round, four values are eliminated, leaving values [3,4,9,6] . In the Final Four round, values [4,9] advance, and eventually 9 is declared the champion.

Tournament

Figure 1-6. A tournament with eight initial values

In this tournament, there are seven less-than invocations (i.e., one for each match), which should be reassuring, since this means the largest value is found with N – 1 uses of less-than, as we had discussed earlier. If you store each of these N – 1 matches, then you can quickly locate the second-largest value, as I now show.

Where can the second-largest value be “hiding” once 9 is declared the champion? Start with 4 as the candidate second-largest value, since this was the value that lost in the Championship round. But the largest value, 9, had two earlier matches, so you must check the other two losing values—value 6 in the Final Four round and value 5 in the Elite Eight round. Thus the second-largest value is 6.

For eight initial values, you need just 2 additional less-than invocations—(is 4 < 6?) and (is 6 < 5?)—to determine that 6 is the second-largest value. It’s no coincidence that 8 = 2 3 and you need 3 – 1 = 2 comparisons. It turns out that for N = 2 K , you need an additional K – 1 comparisons, where K is the number of rounds.

When there are 8 = 2 3 initial values, the algorithm constructs a tournament with 3 rounds. Figure 1-7 visualizes a five-round tournament consisting of 32 values. To double the number of values in the tournament, you only need one additional round of matches; in other words, with each new round K, you can add 2 K more values. Want to find the largest of 64 values? You only need 6 rounds since 2 6 = 64.

Hierarchy

Figure 1-7. A tournament with 32 initial values

To determine the number of rounds for any N, turn to the logarithm function, log() , which is the opposite of the exponent function. With N = 8 initial values, there are 3 rounds required for the tournament, since 2 3 = 8 and log 2 (8) = 3. In this book—and traditionally in computer science—the log() operator is in base 2.

Most handheld calculators compute log() in base 10. The function ln() represents the natural logarithm in base e (which is approximately 2.718). To quickly compute log ( N ) in base 2 using a calculator (or in Microsoft Excel), compute log ( N )/ log ( 2 ).

When N is a power of 2—like 64 or 65,536—the number of rounds in a tournament is log (N), which means the number of additional less-than invocations is log (N) – 1. The algorithm implemented in Listing 1-6 minimizes the invocations of less-than by using extra storage to record the results of all matches.

Listing 1-6. Algorithm to find two largest values in A using tournament

These arrays store the winners and losers of match idx ; there will be N – 1 of them in the tournament.

When a value advances in match m , prior[m] records earlier match, or –1 when it was initial match.

Initialize the first N/2 winner/loser pairs using N/2 invocations of less-than. These represent the matches in the lowest round.

Pair up winners to find a new winner, and record prior match index.

An additional N/2 – 1 invocations of less-than are needed.

Advance m by 2 to find next pair of winners. When idx reaches N – 1, winner[m] is largest.

Initial candidate for second largest, but must check all others that lost to largest to find actual second largest.

No more than log (N) – 1 additional invocations of less-than.

Figure 1-8 shows the execution of this algorithm. After the initialization step, the N values from the original array, A , are separated into N/2 winners and losers ; in the example from Figure 1-6 , there are four pairs. During each advance step in the while loop, the winner/loser of two consecutive matches, m and m+1 , are placed in winner[idx] and loser[idx] respectively; prior[idx] records the prior match from which the winner came (as drawn by an arrow from right to left). After three steps, all match information is stored, and then the algorithm inspects the losers of all prior matches for the winner. You can visualize this by following the arrows backward until they stop. You can see that the candidate second-largest value is found in loser[6] : with just two less-than invocations with loser[5] and loser[2] , it determines which one is largest.

I have just sketched an algorithm to compute the largest and second-largest value in A using just N – 1 + log (N) – 1 = N + log (N) – 2 less-than invocations for any N that is a power of 2. Is tournament_two() practical? Will it outperform largest_two() ? If you only count the number of times less-than is invoked, tournament_two() should be faster. largest_two() requires 131,069 less-than operations on problems of size N = 65,536, while tournament_two() only requires 65,536 + 16 – 2 = 65,550, just about half. But there is more to this story.

Step by step execution

Figure 1-8. Step-by-step execution of tournament algorithm

Table 1-5 reveals that tournament_two() is significantly slower than any of its competitors! Let’s record the total time it takes to solve 100 random problem instances (of size N growing from 1,024 to 2,097,152). While I’m at it, I will include the performance results of the different algorithms from Listing 1-5 . Note that if you run the sample code on your computer, your individual results will be different, but the overall trend in each column will remain the same.

Table 1-5 can be overwhelming to look at, since it just looks like a wall of numbers. If you run these functions on a different computer—perhaps with less memory or a slower CPU—your results might be quite different; however, there are some trends that should reveal themselves no matter on what computer you execute. For the most part, as you read down any column, the time to execute more or less doubles as the problem size doubles.

There are some unexpected situations in this table: note that double_two() starts out being the fastest of the five solutions, but eventually (once N > 262,144), largest_two() becomes the fastest to complete. The clever tournament_two() approach is by far the slowest, simply because it needs to allocate ever-larger storage arrays to be processed. It is so slow, I do not even run it on the largest problem instance because it will take so long.

To make sense of these numbers, Figure 1-9 visualizes the runtime trends as the problem size instance grows ever larger.

Hierarchy

Figure 1-9. Runtime performance comparison

This image reveals more details about the runtime performance of these five approaches:

You can see that the performances of mutable_two() , double_two() , and largest_two() are all more similar to each other than the other two approaches. It’s almost like they are all in the same “family,” all on a straight-line trajectory that appears quite predictable.

tournament_two() is the least efficient, and it noticeably behaves differently from the other approaches. Given that there are so few data points, it is not clear whether it is “curving upward” to greater inefficiencies or whether it also will follow a straight line.

sorting_two() appears to do better than tournament_two() but is slower than the other three approaches. Will it curve further upward, or eventually straighten out?

To understand why these lines are shaped the way they are, you need to learn the two fundamental concepts that explain the inherent complexity of an algorithm.

Time Complexity and Space Complexity

It can be hard to count the number of elementary operations (such as addition, variable assignment, or control logic), because of the difference in programming languages, plus the fact that some languages, such as Java and Python, are executed by an interpreter. But if you could count the total number of elementary operations executed by an algorithm, then you would see that the total number of operations varies based on the size of the problem instance. The goal of time complexity is to come up with a function C (N) that counts the number of elementary operations performed by an algorithm as a function of N, the size of a problem instance.

Assuming that each elementary operation takes a fixed amount of time, t , based on the CPU executing the operation, I can model the time to perform the algorithm as T (N) = t × C (N). Listing 1-7 confirms the insight that the structure of a program is critical. For functions f0 , f1 , f2 , and f3 , you can exactly compute how many times each one executes the operation ct = ct + 1 based on the input size, N. Table 1-6 contains the counts for a few values of N.

Listing 1-7. Four different functions with different performance profiles

The count for f0 is always the same, independent of N. The count for f2 is always seven times greater than f1 , and both of them double in size as N doubles. In contrast, the count for f3 increases far more rapidly; as you have seen before, as N doubles, the count for f3(N) quadruples. Here, f1 and f2 are more similar to each other than they are to f3 . In the next chapter, we will explain the importance of for loops and nested for loops when evaluating an algorithm’s performance.

When evaluating an algorithm, we also have to consider space complexity , which accounts for extra memory required by an algorithm based on the size, N, of a problem instance. Memory is a generic term for data stored in the file system or the RAM of a computer. largest_two() has minimal space requirements: it uses two variables, my_max and second , and an iterator variable, idx . No matter the size of the problem instance, its extra space never changes. This means the space complexity is independent of the size of the problem instance , or constant; mutable_two() has similar behavior. In contrast, tournament_two() allocated three arrays— winner , loser , and prior —all of size N – 1. As N increases, the total extra storage increases in a manner that is directly proportional to the size of the problem instance . 3 Building the tournament structure is going to slow tournament_two() down, when compared against largest_two() . Both double_two() and sorting_two() make a copy of the input, A , which means their storage usage is much more like tournament_two() than largest_two() . Throughout this book, I will evaluate both the time complexity and space complexity of each algorithm.

If you review Table 1-5 , you can see that the timing results for the column largest_two more or less double in subsequent rows; columns double_two and mutable_two behave similarly, as I have already observed. This means that the total time appears to be directly proportional to the size of the problem instance , which is doubling in subsequent rows. This is an important observation, since these functions are more efficient than sorting_two() , which appears to follow a different, less-efficient trajectory. tournament_two() is still the least efficient, with a runtime performance that more than doubles, growing so rapidly that I don’t bother executing it for large problem instances.

As a computer scientist, I cannot just proclaim that the performance curves of largest_two() and mutable_two() “look the same.” I need to rely on a formal theory and notation to capture this idea. In the next chapter, I will present the mathematical tools necessary to analyze algorithm behavior properly.

This chapter provided an introduction to the rich and diverse field of algorithms. I showed how to model the performance of an algorithm on a problem instance of size N by counting the number of key operations it performs. You can also empirically evaluate the runtime performance of the implementation of an algorithm. In both cases, you can determine the order of growth of the algorithm as the problem instance size N doubles.

I introduced several key concepts, including:

Time complexity as estimated by counting the number of key operations executed by an algorithm on a problem instance of size N.

Space complexity as estimated by the amount of memory required when an algorithm executes on a problem instance of size N.

In the next chapter, I will introduce the mathematical tools of asymptotic analysis that will complete my explanation of the techniques needed to properly analyze algorithms.

Challenge Exercises

Palindrome word detector : A palindrome word reads the same backward as forward, such as madam . Devise an algorithm that validates whether a word of N characters is a palindrome. Confirm empirically that it outperforms the two alternatives in Listing 1-8 :

Listing 1-8. Four different functions with different performance profiles

Once you have this problem working, modify it to detect palindrome strings with spaces, punctuation, and mixed capitalization. For example, the following string should classify as a palindrome: " A man, a plan, a canal. Panama! "

Linear time median : A wonderful algorithm exists that efficiently locates the median value in an arbitrary list (for simplicity, assume size of list is odd). Review the code in Listing 1-9 and count the number of times less-than is invoked, using RecordedItem values as shown in the chapter. This implementation rearranges the arbitrary list as it processes.

Listing 1-9. A linear-time algorithm to compute the median value in an unordered list

Implement a different approach (which requires extra storage) that creates a sorted list from the input and selects the middle value. Compare its runtime performance with linear_median() by generating a table of runtime performance.

Counting Sort : If you know that an arbitrary list, A , only contains nonnegative integers from 0 to M, then the following algorithm will properly sort A using just an extra storage of size M.

Listing 1-10 has nested loops—a for loop within a while loop. However, you can demonstrate that A[pos+idx] = v only executes N times.

Listing 1-10. A linear-time Counting Sort algorithm

Conduct a performance analysis to demonstrate that the time to sort N integers in the range from 0 to M doubles as the size of N doubles.

You can eliminate the inner for loop, and improve the performance of this operation, using the ability in Python to replace a sublist using sublist[left:right] = [2,3,4] . Make the change and empirically validate that it, too, doubles as N doubles, while yielding a 30% improvement in speed.

Modify tournament algorithm to work with an odd number of values.

Will the code in Listing 1-11 correctly locate the two largest values in A ?

Listing 1-11. Another attempt to try to compute two largest values in unordered list

Explain the circumstances when this code works correctly and when it fails.

1 The RecordedItem wrapper class overrides the __lt__() less-than operator to count whenever it (or the __gt__() greater-than operator) is invoked.

2 If a match contains two equal values, then only one of these values advances.

3 That is, storage in addition to the data encoding the problem instance, which is not counted as part of the space complexity of any algorithm.

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algorithm method of problem solving

7.3 Problem-Solving

Learning objectives.

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

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

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

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

PROBLEM-SOLVING STRATEGIES

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

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

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

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

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

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

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

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

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

Additional Problem Solving Strategies :

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

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

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

Missionary-Cannibal Problem

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

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

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

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

algorithm method of problem solving

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

algorithm method of problem solving

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

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

GESTALT PSYCHOLOGY AND PROBLEM SOLVING

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

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

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

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

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

Solving puzzles.

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

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

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

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

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

A puzzle involving a scale is shown. At the top of the figure it reads: “Sam Loyds Puzzling Scales.” The first row of the puzzle shows a balanced scale with 3 blocks and a top on the left and 12 marbles on the right. Below this row it reads: “Since the scales now balance.” The next row of the puzzle shows a balanced scale with just the top on the left, and 1 block and 8 marbles on the right. Below this row it reads: “And balance when arranged this way.” The third row shows an unbalanced scale with the top on the left side, which is much lower than the right side. The right side is empty. Below this row it reads: “Then how many marbles will it require to balance with that top?”

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

Pitfalls to problem solving.

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

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

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

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

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

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

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

The first puzzle is a Sudoku grid of 16 squares (4 rows of 4 squares) is shown. Half of the numbers were supplied to start the puzzle and are colored blue, and half have been filled in as the puzzle’s solution and are colored red. The numbers in each row of the grid, left to right, are as follows. Row 1: blue 3, red 1, red 4, blue 2. Row 2: red 2, blue 4, blue 1, red 3. Row 3: red 1, blue 3, blue 2, red 4. Row 4: blue 4, red 2, red 3, blue 1.The second puzzle consists of 9 dots arranged in 3 rows of 3 inside of a square. The solution, four straight lines made without lifting the pencil, is shown in a red line with arrows indicating the direction of movement. In order to solve the puzzle, the lines must extend beyond the borders of the box. The four connecting lines are drawn as follows. Line 1 begins at the top left dot, proceeds through the middle and right dots of the top row, and extends to the right beyond the border of the square. Line 2 extends from the end of line 1, through the right dot of the horizontally centered row, through the middle dot of the bottom row, and beyond the square’s border ending in the space beneath the left dot of the bottom row. Line 3 extends from the end of line 2 upwards through the left dots of the bottom, middle, and top rows. Line 4 extends from the end of line 3 through the middle dot in the middle row and ends at the right dot of the bottom row.

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

References:

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

Review Questions:

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

a. an algorithm

b. a heuristic

c. a mental set

d. trial and error

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

a. divide and conquer

b. means-end analysis

d. experiment

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

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

a. anchoring bias

b. confirmation bias

c. representative bias

d. availability bias

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

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

a. social adjustment

b. student load payment options

c. emotional learning

d. insight learning

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

a. functional fixedness

c. working memory

Critical Thinking Questions:

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

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

Personal Application Question:

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

anchoring bias

availability heuristic

confirmation bias

functional fixedness

hindsight bias

problem-solving strategy

representative bias

trial and error

working backwards

Answers to Exercises

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

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

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

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

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

heuristic:  mental shortcut that saves time when solving a problem

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

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

problem-solving strategy:  method for solving problems

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

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

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

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Chapter 7: Thinking and Intelligence

Problem solving, learning objectives.

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

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

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

PROBLEM-SOLVING STRATEGIES

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

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

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

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

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

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

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

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

A four column by four row Sudoku puzzle is shown. The top left cell contains the number 3. The top right cell contains the number 2. The bottom right cell contains the number 1. The bottom left cell contains the number 4. The cell at the intersection of the second row and the second column contains the number 4. The cell to the right of that contains the number 1. The cell below the cell containing the number 1 contains the number 2. The cell to the left of the cell containing the number 2 contains the number 3.

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

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

A square shaped outline contains three rows and three columns of dots with equal space between them.

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

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

A puzzle involving a scale is shown. At the top of the figure it reads: “Sam Loyds Puzzling Scales.” The first row of the puzzle shows a balanced scale with 3 blocks and a top on the left and 12 marbles on the right. Below this row it reads: “Since the scales now balance.” The next row of the puzzle shows a balanced scale with just the top on the left, and 1 block and 8 marbles on the right. Below this row it reads: “And balance when arranged this way.” The third row shows an unbalanced scale with the top on the left side, which is much lower than the right side. The right side is empty. Below this row it reads: “Then how many marbles will it require to balance with that top?”

PITFALLS TO PROBLEM SOLVING

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

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

Link to Learning

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

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

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

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

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

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

Were you able to determine how many marbles are needed to balance the scales in [link] ? You need nine. Were you able to solve the problems in [link] and [link] ? Here are the answers ( [link] ).

The first puzzle is a Sudoku grid of 16 squares (4 rows of 4 squares) is shown. Half of the numbers were supplied to start the puzzle and are colored blue, and half have been filled in as the puzzle’s solution and are colored red. The numbers in each row of the grid, left to right, are as follows. Row 1:  blue 3, red 1, red 4, blue 2. Row 2: red 2, blue 4, blue 1, red 3. Row 3: red 1, blue 3, blue 2, red 4. Row 4: blue 4, red 2, red 3, blue 1.The second puzzle consists of 9 dots arranged in 3 rows of 3 inside of a square. The solution, four straight lines made without lifting the pencil, is shown in a red line with arrows indicating the direction of movement. In order to solve the puzzle, the lines must extend beyond the borders of the box. The four connecting lines are drawn as follows. Line 1 begins at the top left dot, proceeds through the middle and right dots of the top row, and extends to the right beyond the border of the square. Line 2 extends from the end of line 1, through the right dot of the horizontally centered row, through the middle dot of the bottom row, and beyond the square’s border ending in the space beneath the left dot of the bottom row. Line 3 extends from the end of line 2 upwards through the left dots of the bottom, middle, and top rows. Line 4 extends from the end of line 3 through the middle dot in the middle row and ends at the right dot of the bottom row.

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

Self Check Questions

Critical thinking questions.

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

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

Personal Application Question

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

1. Functional fixedness occurs when you cannot see a use for an object other than the use for which it was intended. For example, if you need something to hold up a tarp in the rain, but only have a pitchfork, you must overcome your expectation that a pitchfork can only be used for garden chores before you realize that you could stick it in the ground and drape the tarp on top of it to hold it up.

2. An algorithm is a proven formula for achieving a desired outcome. It saves time because if you follow it exactly, you will solve the problem without having to figure out how to solve the problem. It is a bit like not reinventing the wheel.

  • Psychology. Authored by : OpenStax College. Located at : http://cnx.org/contents/[email protected]:1/Psychology . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/content/col11629/latest/.

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35 problem-solving techniques and methods for solving complex problems

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

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

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

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

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

Let’s get started! 

How do you identify problems?

How do you identify the right solution.

  • Tips for more effective problem-solving

Complete problem-solving methods

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

Problem-solving warm-up activities

Closing activities for a problem-solving process.

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

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

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

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

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

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

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

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

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

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

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

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

algorithm method of problem solving

Tips for more effective problem solving

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

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

Clearly define the problem

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

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

Don’t jump to conclusions

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

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

Try different approaches  

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

Don’t take it personally 

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

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

Get the right people in the room

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

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

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

Document everything

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

Bring a facilitator 

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

Develop your problem-solving skills

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

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

Design a great agenda

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

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

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

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

  • Six Thinking Hats
  • Lightning Decision Jam
  • Problem Definition Process
  • Discovery & Action Dialogue
Design Sprint 2.0
  • Open Space Technology

1. Six Thinking Hats

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

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

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

2. Lightning Decision Jam

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

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

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

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

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

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

3. Problem Definition Process

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

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

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

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

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

4. The 5 Whys 

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

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

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

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

5. World Cafe

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

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

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

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

6. Discovery & Action Dialogue (DAD)

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

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

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

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

7. Design Sprint 2.0

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

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

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

8. Open space technology

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

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

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

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

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

Techniques to identify and analyze problems

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

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

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

Let’s take a look!

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

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

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

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

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

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

10. The Creativity Dice

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

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

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

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

11. Fishbone Analysis

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

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

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

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

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

12. Problem Tree 

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

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

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

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

13. SWOT Analysis

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

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

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

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

14. Agreement-Certainty Matrix

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

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

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

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

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

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

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

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

16. Speed Boat

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

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

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

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

17. The Journalistic Six

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

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

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

18. LEGO Challenge

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

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

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

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

19. What, So What, Now What?

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

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

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

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

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

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

20. Journalists  

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

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

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

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

Problem-solving techniques for developing solutions 

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

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

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

21. Mindspin  

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

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

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

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

22. Improved Solutions

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

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

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

23. Four Step Sketch

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

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

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

24. 15% Solutions

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

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

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

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

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

25. How-Now-Wow Matrix

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

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

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

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

26. Impact and Effort Matrix

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

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

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

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

27. Dotmocracy

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

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

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

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

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

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

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

28. Check-in / Check-out

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

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

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

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

29. Doodling Together  

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

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

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

30. Show and Tell

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

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

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

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

31. Constellations

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

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

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

32. Draw a Tree

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

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

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

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

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

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

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

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

How do I conclude a problem-solving process?

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

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

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

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

33. One Breath Feedback

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

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

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

34. Who What When Matrix 

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

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

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

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

35. Response cards

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

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

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

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

Save time and effort discovering the right solutions

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

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

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

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

algorithm method of problem solving

Over to you

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

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

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  • Published: 10 April 2024

A hybrid particle swarm optimization algorithm for solving engineering problem

  • Jinwei Qiao 1 , 2 ,
  • Guangyuan Wang 1 , 2 ,
  • Zhi Yang 1 , 2 ,
  • Xiaochuan Luo 3 ,
  • Jun Chen 1 , 2 ,
  • Kan Li 4 &
  • Pengbo Liu 1 , 2  

Scientific Reports volume  14 , Article number:  8357 ( 2024 ) Cite this article

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  • Computational science
  • Mechanical engineering

To overcome the disadvantages of premature convergence and easy trapping into local optimum solutions, this paper proposes an improved particle swarm optimization algorithm (named NDWPSO algorithm) based on multiple hybrid strategies. Firstly, the elite opposition-based learning method is utilized to initialize the particle position matrix. Secondly, the dynamic inertial weight parameters are given to improve the global search speed in the early iterative phase. Thirdly, a new local optimal jump-out strategy is proposed to overcome the "premature" problem. Finally, the algorithm applies the spiral shrinkage search strategy from the whale optimization algorithm (WOA) and the Differential Evolution (DE) mutation strategy in the later iteration to accelerate the convergence speed. The NDWPSO is further compared with other 8 well-known nature-inspired algorithms (3 PSO variants and 5 other intelligent algorithms) on 23 benchmark test functions and three practical engineering problems. Simulation results prove that the NDWPSO algorithm obtains better results for all 49 sets of data than the other 3 PSO variants. Compared with 5 other intelligent algorithms, the NDWPSO obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( \({f}_{1}-{f}_{13}\) ) with 3 kinds of dimensional spaces (Dim = 30,50,100) and 80% of the best optimal solutions for 10 fixed-multimodal benchmark functions. Also, the best design solutions are obtained by NDWPSO for all 3 classical practical engineering problems.

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Introduction

In the ever-changing society, new optimization problems arise every moment, and they are distributed in various fields, such as automation control 1 , statistical physics 2 , security prevention and temperature prediction 3 , artificial intelligence 4 , and telecommunication technology 5 . Faced with a constant stream of practical engineering optimization problems, traditional solution methods gradually lose their efficiency and convenience, making it more and more expensive to solve the problems. Therefore, researchers have developed many metaheuristic algorithms and successfully applied them to the solution of optimization problems. Among them, Particle swarm optimization (PSO) algorithm 6 is one of the most widely used swarm intelligence algorithms.

However, the basic PSO has a simple operating principle and solves problems with high efficiency and good computational performance, but it suffers from the disadvantages of easily trapping in local optima and premature convergence. To improve the overall performance of the particle swarm algorithm, an improved particle swarm optimization algorithm is proposed by the multiple hybrid strategy in this paper. The improved PSO incorporates the search ideas of other intelligent algorithms (DE, WOA), so the improved algorithm proposed in this paper is named NDWPSO. The main improvement schemes are divided into the following 4 points: Firstly, a strategy of elite opposition-based learning is introduced into the particle population position initialization. A high-quality initialization matrix of population position can improve the convergence speed of the algorithm. Secondly, a dynamic weight methodology is adopted for the acceleration coefficients by combining the iterative map and linearly transformed method. This method utilizes the chaotic nature of the mapping function, the fast convergence capability of the dynamic weighting scheme, and the time-varying property of the acceleration coefficients. Thus, the global search and local search of the algorithm are balanced and the global search speed of the population is improved. Thirdly, a determination mechanism is set up to detect whether the algorithm falls into a local optimum. When the algorithm is “premature”, the population resets 40% of the position information to overcome the local optimum. Finally, the spiral shrinking mechanism combined with the DE/best/2 position mutation is used in the later iteration, which further improves the solution accuracy.

The structure of the paper is given as follows: Sect. “ Particle swarm optimization (PSO) ” describes the principle of the particle swarm algorithm. Section “ Improved particle swarm optimization algorithm ” shows the detailed improvement strategy and a comparison experiment of inertia weight is set up for the proposed NDWPSO. Section “ Experiment and discussion ” includes the experimental and result discussion sections on the performance of the improved algorithm. Section “ Conclusions and future works ” summarizes the main findings of this study.

Literature review

This section reviews some metaheuristic algorithms and other improved PSO algorithms. A simple discussion about recently proposed research studies is given.

Metaheuristic algorithms

A series of metaheuristic algorithms have been proposed in recent years by using various innovative approaches. For instance, Lin et al. 7 proposed a novel artificial bee colony algorithm (ABCLGII) in 2018 and compared ABCLGII with other outstanding ABC variants on 52 frequently used test functions. Abed-alguni et al. 8 proposed an exploratory cuckoo search (ECS) algorithm in 2021 and carried out several experiments to investigate the performance of ECS by 14 benchmark functions. Brajević 9 presented a novel shuffle-based artificial bee colony (SB-ABC) algorithm for solving integer programming and minimax problems in 2021. The experiments are tested on 7 integer programming problems and 10 minimax problems. In 2022, Khan et al. 10 proposed a non-deterministic meta-heuristic algorithm called Non-linear Activated Beetle Antennae Search (NABAS) for a non-convex tax-aware portfolio selection problem. Brajević et al. 11 proposed a hybridization of the sine cosine algorithm (HSCA) in 2022 to solve 15 complex structural and mechanical engineering design optimization problems. Abed-Alguni et al. 12 proposed an improved Salp Swarm Algorithm (ISSA) in 2022 for single-objective continuous optimization problems. A set of 14 standard benchmark functions was used to evaluate the performance of ISSA. In 2023, Nadimi et al. 13 proposed a binary starling murmuration optimization (BSMO) to select the effective features from different important diseases. In the same year, Nadimi et al. 14 systematically reviewed the last 5 years' developments of WOA and made a critical analysis of those WOA variants. In 2024, Fatahi et al. 15 proposed an Improved Binary Quantum-based Avian Navigation Optimizer Algorithm (IBQANA) for the Feature Subset Selection problem in the medical area. Experimental evaluation on 12 medical datasets demonstrates that IBQANA outperforms 7 established algorithms. Abed-alguni et al. 16 proposed an Improved Binary DJaya Algorithm (IBJA) to solve the Feature Selection problem in 2024. The IBJA’s performance was compared against 4 ML classifiers and 10 efficient optimization algorithms.

Improved PSO algorithms

Many researchers have constantly proposed some improved PSO algorithms to solve engineering problems in different fields. For instance, Yeh 17 proposed an improved particle swarm algorithm, which combines a new self-boundary search and a bivariate update mechanism, to solve the reliability redundancy allocation problem (RRAP) problem. Solomon et al. 18 designed a collaborative multi-group particle swarm algorithm with high parallelism that was used to test the adaptability of Graphics Processing Units (GPUs) in distributed computing environments. Mukhopadhyay and Banerjee 19 proposed a chaotic multi-group particle swarm optimization (CMS-PSO) to estimate the unknown parameters of an autonomous chaotic laser system. Duan et al. 20 designed an improved particle swarm algorithm with nonlinear adjustment of inertia weights to improve the coupling accuracy between laser diodes and single-mode fibers. Sun et al. 21 proposed a particle swarm optimization algorithm combined with non-Gaussian stochastic distribution for the optimal design of wind turbine blades. Based on a multiple swarm scheme, Liu et al. 22 proposed an improved particle swarm optimization algorithm to predict the temperatures of steel billets for the reheating furnace. In 2022, Gad 23 analyzed the existing 2140 papers on Swarm Intelligence between 2017 and 2019 and pointed out that the PSO algorithm still needs further research. In general, the improved methods can be classified into four categories:

Adjusting the distribution of algorithm parameters. Feng et al. 24 used a nonlinear adaptive method on inertia weights to balance local and global search and introduced asynchronously varying acceleration coefficients.

Changing the updating formula of the particle swarm position. Both papers 25 and 26 used chaotic mapping functions to update the inertia weight parameters and combined them with a dynamic weighting strategy to update the particle swarm positions. This improved approach enables the particle swarm algorithm to be equipped with fast convergence of performance.

The initialization of the swarm. Alsaidy and Abbood proposed 27 a hybrid task scheduling algorithm that replaced the random initialization of the meta-heuristic algorithm with the heuristic algorithms MCT-PSO and LJFP-PSO.

Combining with other intelligent algorithms: Liu et al. 28 introduced the differential evolution (DE) algorithm into PSO to increase the particle swarm as diversity and reduce the probability of the population falling into local optimum.

Particle swarm optimization (PSO)

The particle swarm optimization algorithm is a population intelligence algorithm for solving continuous and discrete optimization problems. It originated from the social behavior of individuals in bird and fish flocks 6 . The core of the PSO algorithm is that an individual particle identifies potential solutions by flight in a defined constraint space adjusts its exploration direction to approach the global optimal solution based on the shared information among the group, and finally solves the optimization problem. Each particle \(i\) includes two attributes: velocity vector \({V}_{i}=\left[{v}_{i1},{v}_{i2},{v}_{i3},{...,v}_{ij},{...,v}_{iD},\right]\) and position vector \({X}_{i}=[{x}_{i1},{x}_{i2},{x}_{i3},...,{x}_{ij},...,{x}_{iD}]\) . The velocity vector is used to modify the motion path of the swarm; the position vector represents a potential solution for the optimization problem. Here, \(j=\mathrm{1,2},\dots ,D\) , \(D\) represents the dimension of the constraint space. The equations for updating the velocity and position of the particle swarm are shown in Eqs. ( 1 ) and ( 2 ).

Here \({Pbest}_{i}^{k}\) represents the previous optimal position of the particle \(i\) , and \({Gbest}\) is the optimal position discovered by the whole population. \(i=\mathrm{1,2},\dots ,n\) , \(n\) denotes the size of the particle swarm. \({c}_{1}\) and \({c}_{2}\) are the acceleration constants, which are used to adjust the search step of the particle 29 . \({r}_{1}\) and \({r}_{2}\) are two random uniform values distributed in the range \([\mathrm{0,1}]\) , which are used to improve the randomness of the particle search. \(\omega\) inertia weight parameter, which is used to adjust the scale of the search range of the particle swarm 30 . The basic PSO sets the inertia weight parameter as a time-varying parameter to balance global exploration and local seeking. The updated equation of the inertia weight parameter is given as follows:

where \({\omega }_{max}\) and \({\omega }_{min}\) represent the upper and lower limits of the range of inertia weight parameter. \(k\) and \(Mk\) are the current iteration and maximum iteration.

Improved particle swarm optimization algorithm

According to the no free lunch theory 31 , it is known that no algorithm can solve every practical problem with high quality and efficiency for increasingly complex and diverse optimization problems. In this section, several improvement strategies are proposed to improve the search efficiency and overcome this shortcoming of the basic PSO algorithm.

Improvement strategies

The optimization strategies of the improved PSO algorithm are shown as follows:

The inertia weight parameter is updated by an improved chaotic variables method instead of a linear decreasing strategy. Chaotic mapping performs the whole search at a higher speed and is more resistant to falling into local optimal than the probability-dependent random search 32 . However, the population may result in that particles can easily fly out of the global optimum boundary. To ensure that the population can converge to the global optimum, an improved Iterative mapping is adopted and shown as follows:

Here \({\omega }_{k}\) is the inertia weight parameter in the iteration \(k\) , \(b\) is the control parameter in the range \([\mathrm{0,1}]\) .

The acceleration coefficients are updated by the linear transformation. \({c}_{1}\) and \({c}_{2}\) represent the influential coefficients of the particles by their own and population information, respectively. To improve the search performance of the population, \({c}_{1}\) and \({c}_{2}\) are changed from fixed values to time-varying parameter parameters, that are updated by linear transformation with the number of iterations:

where \({c}_{max}\) and \({c}_{min}\) are the maximum and minimum values of acceleration coefficients, respectively.

The initialization scheme is determined by elite opposition-based learning . The high-quality initial population will accelerate the solution speed of the algorithm and improve the accuracy of the optimal solution. Thus, the elite backward learning strategy 33 is introduced to generate the position matrix of the initial population. Suppose the elite individual of the population is \({X}=[{x}_{1},{x}_{2},{x}_{3},...,{x}_{j},...,{x}_{D}]\) , and the elite opposition-based solution of \(X\) is \({X}_{o}=[{x}_{{\text{o}}1},{x}_{{\text{o}}2},{x}_{{\text{o}}3},...,{x}_{oj},...,{x}_{oD}]\) . The formula for the elite opposition-based solution is as follows:

where \({k}_{r}\) is the random value in the range \((\mathrm{0,1})\) . \({ux}_{oij}\) and \({lx}_{oij}\) are dynamic boundaries of the elite opposition-based solution in \(j\) dimensional variables. The advantage of dynamic boundary is to reduce the exploration space of particles, which is beneficial to the convergence of the algorithm. When the elite opposition-based solution is out of bounds, the out-of-bounds processing is performed. The equation is given as follows:

After calculating the fitness function values of the elite solution and the elite opposition-based solution, respectively, \(n\) high quality solutions were selected to form a new initial population position matrix.

The position updating Eq. ( 2 ) is modified based on the strategy of dynamic weight. To improve the speed of the global search of the population, the strategy of dynamic weight from the artificial bee colony algorithm 34 is introduced to enhance the computational performance. The new position updating equation is shown as follows:

Here \(\rho\) is the random value in the range \((\mathrm{0,1})\) . \(\psi\) represents the acceleration coefficient and \({\omega }{\prime}\) is the dynamic weight coefficient. The updated equations of the above parameters are as follows:

where \(f(i)\) denotes the fitness function value of individual particle \(i\) and u is the average of the population fitness function values in the current iteration. The Eqs. ( 11 , 12 ) are introduced into the position updating equation. And they can attract the particle towards positions of the best-so-far solution in the search space.

New local optimal jump-out strategy is added for escaping from the local optimal. When the value of the fitness function for the population optimal particles does not change in M iterations, the algorithm determines that the population falls into a local optimal. The scheme in which the population jumps out of the local optimum is to reset the position information of the 40% of individuals within the population, in other words, to randomly generate the position vector in the search space. M is set to 5% of the maximum number of iterations.

New spiral update search strategy is added after the local optimal jump-out strategy. Since the whale optimization algorithm (WOA) was good at exploring the local search space 35 , the spiral update search strategy in the WOA 36 is introduced to update the position of the particles after the swarm jumps out of local optimal. The equation for the spiral update is as follows:

Here \(D=\left|{x}_{i}\left(k\right)-Gbest\right|\) denotes the distance between the particle itself and the global optimal solution so far. \(B\) is the constant that defines the shape of the logarithmic spiral. \(l\) is the random value in \([-\mathrm{1,1}]\) . \(l\) represents the distance between the newly generated particle and the global optimal position, \(l=-1\) means the closest distance, while \(l=1\) means the farthest distance, and the meaning of this parameter can be directly observed by Fig.  1 .

figure 1

Spiral updating position.

The DE/best/2 mutation strategy is introduced to form the mutant particle. 4 individuals in the population are randomly selected that differ from the current particle, then the vector difference between them is rescaled, and the difference vector is combined with the global optimal position to form the mutant particle. The equation for mutation of particle position is shown as follows:

where \({x}^{*}\) is the mutated particle, \(F\) is the scale factor of mutation, \({r}_{1}\) , \({r}_{2}\) , \({r}_{3}\) , \({r}_{4}\) are random integer values in \((0,n]\) and not equal to \(i\) , respectively. Specific particles are selected for mutation with the screening conditions as follows:

where \(Cr\) represents the probability of mutation, \(rand\left(\mathrm{0,1}\right)\) is a random number in \(\left(\mathrm{0,1}\right)\) , and \({i}_{rand}\) is a random integer value in \((0,n]\) .

The improved PSO incorporates the search ideas of other intelligent algorithms (DE, WOA), so the improved algorithm proposed in this paper is named NDWPSO. The pseudo-code for the NDWPSO algorithm is given as follows:

figure a

The main procedure of NDWPSO.

Comparing the distribution of inertia weight parameters

There are several improved PSO algorithms (such as CDWPSO 25 , and SDWPSO 26 ) that adopt the dynamic weighted particle position update strategy as their improvement strategy. The updated equations of the CDWPSO and the SDWPSO algorithm for the inertia weight parameters are given as follows:

where \({\text{A}}\) is a value in \((\mathrm{0,1}]\) . \({r}_{max}\) and \({r}_{min}\) are the upper and lower limits of the fluctuation range of the inertia weight parameters, \(k\) is the current number of algorithm iterations, and \(Mk\) denotes the maximum number of iterations.

Considering that the update method of inertia weight parameters by our proposed NDWPSO is comparable to the CDWPSO, and SDWPSO, a comparison experiment for the distribution of inertia weight parameters is set up in this section. The maximum number of iterations in the experiment is \(Mk=500\) . The distributions of CDWPSO, SDWPSO, and NDWPSO inertia weights are shown sequentially in Fig.  2 .

figure 2

The inertial weight distribution of CDWPSO, SDWPSO, and NDWPSO.

In Fig.  2 , the inertia weight value of CDWPSO is a random value in (0,1]. It may make individual particles fly out of the range in the late iteration of the algorithm. Similarly, the inertia weight value of SDWPSO is a value that tends to zero infinitely, so that the swarm no longer can fly in the search space, making the algorithm extremely easy to fall into the local optimal value. On the other hand, the distribution of the inertia weights of the NDWPSO forms a gentle slope by two curves. Thus, the swarm can faster lock the global optimum range in the early iterations and locate the global optimal more precisely in the late iterations. The reason is that the inertia weight values between two adjacent iterations are inversely proportional to each other. Besides, the time-varying part of the inertial weight within NDWPSO is designed to reduce the chaos characteristic of the parameters. The inertia weight value of NDWPSO avoids the disadvantages of the above two schemes, so its design is more reasonable.

Experiment and discussion

In this section, three experiments are set up to evaluate the performance of NDWPSO: (1) the experiment of 23 classical functions 37 between NDWPSO and three particle swarm algorithms (PSO 6 , CDWPSO 25 , SDWPSO 26 ); (2) the experiment of benchmark test functions between NDWPSO and other intelligent algorithms (Whale Optimization Algorithm (WOA) 36 , Harris Hawk Algorithm (HHO) 38 , Gray Wolf Optimization Algorithm (GWO) 39 , Archimedes Algorithm (AOA) 40 , Equilibrium Optimizer (EO) 41 and Differential Evolution (DE) 42 ); (3) the experiment for solving three real engineering problems (welded beam design 43 , pressure vessel design 44 , and three-bar truss design 38 ). All experiments are run on a computer with Intel i5-11400F GPU, 2.60 GHz, 16 GB RAM, and the code is written with MATLAB R2017b.

The benchmark test functions are 23 classical functions, which consist of indefinite unimodal (F1–F7), indefinite dimensional multimodal functions (F8–F13), and fixed-dimensional multimodal functions (F14–F23). The unimodal benchmark function is used to evaluate the global search performance of different algorithms, while the multimodal benchmark function reflects the ability of the algorithm to escape from the local optimal. The mathematical equations of the benchmark functions are shown and found as Supplementary Tables S1 – S3 online.

Experiments on benchmark functions between NDWPSO, and other PSO variants

The purpose of the experiment is to show the performance advantages of the NDWPSO algorithm. Here, the dimensions and corresponding population sizes of 13 benchmark functions (7 unimodal and 6 multimodal) are set to (30, 40), (50, 70), and (100, 130). The population size of 10 fixed multimodal functions is set to 40. Each algorithm is repeated 30 times independently, and the maximum number of iterations is 200. The performance of the algorithm is measured by the mean and the standard deviation (SD) of the results for different benchmark functions. The parameters of the NDWPSO are set as: \({[{\omega }_{min},\omega }_{max}]=[\mathrm{0.4,0.9}]\) , \(\left[{c}_{max},{c}_{min}\right]=\left[\mathrm{2.5,1.5}\right],{V}_{max}=0.1,b={e}^{-50}, M=0.05\times Mk, B=1,F=0.7, Cr=0.9.\) And, \(A={\omega }_{max}\) for CDWPSO; \({[r}_{max},{r}_{min}]=[\mathrm{4,0}]\) for SDWPSO.

Besides, the experimental data are retained to two decimal places, but some experimental data will increase the number of retained data to pursue more accuracy in comparison. The best results in each group of experiments will be displayed in bold font. The experimental data is set to 0 if the value is below 10 –323 . The experimental parameter settings in this paper are different from the references (PSO 6 , CDWPSO 25 , SDWPSO 26 , so the final experimental data differ from the ones within the reference.

As shown in Tables 1 and 2 , the NDWPSO algorithm obtains better results for all 49 sets of data than other PSO variants, which include not only 13 indefinite-dimensional benchmark functions and 10 fixed-multimodal benchmark functions. Remarkably, the SDWPSO algorithm obtains the same accuracy of calculation as NDWPSO for both unimodal functions f 1 –f 4 and multimodal functions f 9 –f 11 . The solution accuracy of NDWPSO is higher than that of other PSO variants for fixed-multimodal benchmark functions f 14 -f 23 . The conclusion can be drawn that the NDWPSO has excellent global search capability, local search capability, and the capability for escaping the local optimal.

In addition, the convergence curves of the 23 benchmark functions are shown in Figs. 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 and 19 . The NDWPSO algorithm has a faster convergence speed in the early stage of the search for processing functions f1-f6, f8-f14, f16, f17, and finds the global optimal solution with a smaller number of iterations. In the remaining benchmark function experiments, the NDWPSO algorithm shows no outstanding performance for convergence speed in the early iterations. There are two reasons of no outstanding performance in the early iterations. On one hand, the fixed-multimodal benchmark function has many disturbances and local optimal solutions in the whole search space. on the other hand, the initialization scheme based on elite opposition-based learning is still stochastic, which leads to the initial position far from the global optimal solution. The inertia weight based on chaotic mapping and the strategy of spiral updating can significantly improve the convergence speed and computational accuracy of the algorithm in the late search stage. Finally, the NDWPSO algorithm can find better solutions than other algorithms in the middle and late stages of the search.

figure 3

Evolution curve of NDWPSO and other PSO algorithms for f1 (Dim = 30,50,100).

figure 4

Evolution curve of NDWPSO and other PSO algorithms for f2 (Dim = 30,50,100).

figure 5

Evolution curve of NDWPSO and other PSO algorithms for f3 (Dim = 30,50,100).

figure 6

Evolution curve of NDWPSO and other PSO algorithms for f4 (Dim = 30,50,100).

figure 7

Evolution curve of NDWPSO and other PSO algorithms for f5 (Dim = 30,50,100).

figure 8

Evolution curve of NDWPSO and other PSO algorithms for f6 (Dim = 30,50,100).

figure 9

Evolution curve of NDWPSO and other PSO algorithms for f7 (Dim = 30,50,100).

figure 10

Evolution curve of NDWPSO and other PSO algorithms for f8 (Dim = 30,50,100).

figure 11

Evolution curve of NDWPSO and other PSO algorithms for f9 (Dim = 30,50,100).

figure 12

Evolution curve of NDWPSO and other PSO algorithms for f10 (Dim = 30,50,100).

figure 13

Evolution curve of NDWPSO and other PSO algorithms for f11(Dim = 30,50,100).

figure 14

Evolution curve of NDWPSO and other PSO algorithms for f12 (Dim = 30,50,100).

figure 15

Evolution curve of NDWPSO and other PSO algorithms for f13 (Dim = 30,50,100).

figure 16

Evolution curve of NDWPSO and other PSO algorithms for f14, f15, f16.

figure 17

Evolution curve of NDWPSO and other PSO algorithms for f17, f18, f19.

figure 18

Evolution curve of NDWPSO and other PSO algorithms for f20, f21, f22.

figure 19

Evolution curve of NDWPSO and other PSO algorithms for f23.

To evaluate the performance of different PSO algorithms, a statistical test is conducted. Due to the stochastic nature of the meta-heuristics, it is not enough to compare algorithms based on only the mean and standard deviation values. The optimization results cannot be assumed to obey the normal distribution; thus, it is necessary to judge whether the results of the algorithms differ from each other in a statistically significant way. Here, the Wilcoxon non-parametric statistical test 45 is used to obtain a parameter called p -value to verify whether two sets of solutions are different to a statistically significant extent or not. Generally, it is considered that p  ≤ 0.5 can be considered as a statistically significant superiority of the results. The p -values calculated in Wilcoxon’s rank-sum test comparing NDWPSO and other PSO algorithms are listed in Table  3 for all benchmark functions. The p -values in Table  3 additionally present the superiority of the NDWPSO because all of the p -values are much smaller than 0.5.

In general, the NDWPSO has the fastest convergence rate when finding the global optimum from Figs. 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 and 19 , and thus we can conclude that the NDWPSO is superior to the other PSO variants during the process of optimization.

Comparison experiments between NDWPSO and other intelligent algorithms

Experiments are conducted to compare NDWPSO with several other intelligent algorithms (WOA, HHO, GWO, AOA, EO and DE). The experimental object is 23 benchmark functions, and the experimental parameters of the NDWPSO algorithm are set the same as in Experiment 4.1. The maximum number of iterations of the experiment is increased to 2000 to fully demonstrate the performance of each algorithm. Each algorithm is repeated 30 times individually. The parameters of the relevant intelligent algorithms in the experiments are set as shown in Table 4 . To ensure the fairness of the algorithm comparison, all parameters are concerning the original parameters in the relevant algorithm literature. The experimental results are shown in Tables 5 , 6 , 7 and 8 and Figs. 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 and 36 .

figure 20

Evolution curve of NDWPSO and other algorithms for f1 (Dim = 30,50,100).

figure 21

Evolution curve of NDWPSO and other algorithms for f2 (Dim = 30,50,100).

figure 22

Evolution curve of NDWPSO and other algorithms for f3(Dim = 30,50,100).

figure 23

Evolution curve of NDWPSO and other algorithms for f4 (Dim = 30,50,100).

figure 24

Evolution curve of NDWPSO and other algorithms for f5 (Dim = 30,50,100).

figure 25

Evolution curve of NDWPSO and other algorithms for f6 (Dim = 30,50,100).

figure 26

Evolution curve of NDWPSO and other algorithms for f7 (Dim = 30,50,100).

figure 27

Evolution curve of NDWPSO and other algorithms for f8 (Dim = 30,50,100).

figure 28

Evolution curve of NDWPSO and other algorithms for f9(Dim = 30,50,100).

figure 29

Evolution curve of NDWPSO and other algorithms for f10 (Dim = 30,50,100).

figure 30

Evolution curve of NDWPSO and other algorithms for f11 (Dim = 30,50,100).

figure 31

Evolution curve of NDWPSO and other algorithms for f12 (Dim = 30,50,100).

figure 32

Evolution curve of NDWPSO and other algorithms for f13 (Dim = 30,50,100).

figure 33

Evolution curve of NDWPSO and other algorithms for f14, f15, f16.

figure 34

Evolution curve of NDWPSO and other algorithms for f17, f18, f19.

figure 35

Evolution curve of NDWPSO and other algorithms for f20, f21, f22.

figure 36

Evolution curve of NDWPSO and other algorithms for f23.

The experimental data of NDWPSO and other intelligent algorithms for handling 30, 50, and 100-dimensional benchmark functions ( \({f}_{1}-{f}_{13}\) ) are recorded in Tables 8 , 9 and 10 , respectively. The comparison data of fixed-multimodal benchmark tests ( \({f}_{14}-{f}_{23}\) ) are recorded in Table 11 . According to the data in Tables 5 , 6 and 7 , the NDWPSO algorithm obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( \({f}_{1}-{f}_{13}\) ) in the search space of three dimensions (Dim = 30, 50, 100), respectively. In Table 8 , the NDWPSO algorithm obtains 80% of the optimal solutions in 10 fixed-multimodal benchmark functions.

The convergence curves of each algorithm are shown in Figs. 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 and 36 . The NDWPSO algorithm demonstrates two convergence behaviors when calculating the benchmark functions in 30, 50, and 100-dimensional search spaces. The first behavior is the fast convergence of NDWPSO with a small number of iterations at the beginning of the search. The reason is that the Iterative-mapping strategy and the position update scheme of dynamic weighting are used in the NDWPSO algorithm. This scheme can quickly target the region in the search space where the global optimum is located, and then precisely lock the optimal solution. When NDWPSO processes the functions \({f}_{1}-{f}_{4}\) , and \({f}_{9}-{f}_{11}\) , the behavior can be reflected in the convergence trend of their corresponding curves. The second behavior is that NDWPSO gradually improves the convergence accuracy and rapidly approaches the global optimal in the middle and late stages of the iteration. The NDWPSO algorithm fails to converge quickly in the early iterations, which is possible to prevent the swarm from falling into a local optimal. The behavior can be demonstrated by the convergence trend of the curves when NDWPSO handles the functions \({f}_{6}\) , \({f}_{12}\) , and \({f}_{13}\) , and it also shows that the NDWPSO algorithm has an excellent ability of local search.

Combining the experimental data with the convergence curves, it is concluded that the NDWPSO algorithm has a faster convergence speed, so the effectiveness and global convergence of the NDWPSO algorithm are more outstanding than other intelligent algorithms.

Experiments on classical engineering problems

Three constrained classical engineering design problems (welded beam design, pressure vessel design 43 , and three-bar truss design 38 ) are used to evaluate the NDWPSO algorithm. The experiments are the NDWPSO algorithm and 5 other intelligent algorithms (WOA 36 , HHO, GWO, AOA, EO 41 ). Each algorithm is provided with the maximum number of iterations and population size ( \({\text{Mk}}=500,\mathrm{ n}=40\) ), and then repeats 30 times, independently. The parameters of the algorithms are set the same as in Table 4 . The experimental results of three engineering design problems are recorded in Tables 9 , 10 and 11 in turn. The result data is the average value of the solved data.

Welded beam design

The target of the welded beam design problem is to find the optimal manufacturing cost for the welded beam with the constraints, as shown in Fig.  37 . The constraints are the thickness of the weld seam ( \({\text{h}}\) ), the length of the clamped bar ( \({\text{l}}\) ), the height of the bar ( \({\text{t}}\) ) and the thickness of the bar ( \({\text{b}}\) ). The mathematical formulation of the optimization problem is given as follows:

figure 37

Welded beam design.

In Table 9 , the NDWPSO, GWO, and EO algorithms obtain the best optimal cost. Besides, the standard deviation (SD) of t NDWPSO is the lowest, which means it has very good results in solving the welded beam design problem.

Pressure vessel design

Kannan and Kramer 43 proposed the pressure vessel design problem as shown in Fig.  38 to minimize the total cost, including the cost of material, forming, and welding. There are four design optimized objects: the thickness of the shell \({T}_{s}\) ; the thickness of the head \({T}_{h}\) ; the inner radius \({\text{R}}\) ; the length of the cylindrical section without considering the head \({\text{L}}\) . The problem includes the objective function and constraints as follows:

figure 38

Pressure vessel design.

The results in Table 10 show that the NDWPSO algorithm obtains the lowest optimal cost with the same constraints and has the lowest standard deviation compared with other algorithms, which again proves the good performance of NDWPSO in terms of solution accuracy.

Three-bar truss design

This structural design problem 44 is one of the most widely-used case studies as shown in Fig.  39 . There are two main design parameters: the area of the bar1 and 3 ( \({A}_{1}={A}_{3}\) ) and area of bar 2 ( \({A}_{2}\) ). The objective is to minimize the weight of the truss. This problem is subject to several constraints as well: stress, deflection, and buckling constraints. The problem is formulated as follows:

figure 39

Three-bar truss design.

From Table 11 , NDWPSO obtains the best design solution in this engineering problem and has the smallest standard deviation of the result data. In summary, the NDWPSO can reveal very competitive results compared to other intelligent algorithms.

Conclusions and future works

An improved algorithm named NDWPSO is proposed to enhance the solving speed and improve the computational accuracy at the same time. The improved NDWPSO algorithm incorporates the search ideas of other intelligent algorithms (DE, WOA). Besides, we also proposed some new hybrid strategies to adjust the distribution of algorithm parameters (such as the inertia weight parameter, the acceleration coefficients, the initialization scheme, the position updating equation, and so on).

23 classical benchmark functions: indefinite unimodal (f1-f7), indefinite multimodal (f8-f13), and fixed-dimensional multimodal(f14-f23) are applied to evaluate the effective line and feasibility of the NDWPSO algorithm. Firstly, NDWPSO is compared with PSO, CDWPSO, and SDWPSO. The simulation results can prove the exploitative, exploratory, and local optima avoidance of NDWPSO. Secondly, the NDWPSO algorithm is compared with 5 other intelligent algorithms (WOA, HHO, GWO, AOA, EO). The NDWPSO algorithm also has better performance than other intelligent algorithms. Finally, 3 classical engineering problems are applied to prove that the NDWPSO algorithm shows superior results compared to other algorithms for the constrained engineering optimization problems.

Although the proposed NDWPSO is superior in many computation aspects, there are still some limitations and further improvements are needed. The NDWPSO performs a limit initialize on each particle by the strategy of “elite opposition-based learning”, it takes more computation time before speed update. Besides, the” local optimal jump-out” strategy also brings some random process. How to reduce the random process and how to improve the limit initialize efficiency are the issues that need to be further discussed. In addition, in future work, researchers will try to apply the NDWPSO algorithm to wider fields to solve more complex and diverse optimization problems.

Data availability

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

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Acknowledgements

This work was supported by Key R&D plan of Shandong Province, China (2021CXGC010207, 2023CXGC01020); First batch of talent research projects of Qilu University of Technology in 2023 (2023RCKY116); Introduction of urgently needed talent projects in Key Supported Regions of Shandong Province; Key Projects of Natural Science Foundation of Shandong Province (ZR2020ME116); the Innovation Ability Improvement Project for Technology-based Small- and Medium-sized Enterprises of Shandong Province (2022TSGC2051, 2023TSGC0024, 2023TSGC0931); National Key R&D Program of China (2019YFB1705002), LiaoNing Revitalization Talents Program (XLYC2002041) and Young Innovative Talents Introduction & Cultivation Program for Colleges and Universities of Shandong Province (Granted by Department of Education of Shandong Province, Sub-Title: Innovative Research Team of High Performance Integrated Device).

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Research Article

A novel and effective method for solving the router nodes placement in wireless mesh networks using reinforcement learning

Contributed equally to this work with: Le Huu Binh, Thuy-Van T. Duong

Roles Investigation, Methodology, Software, Writing – original draft, Writing – review & editing

Affiliation Faculty of Information Technology, University of Sciences, Hue University, Hue City, Vietnam

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* E-mail: [email protected]

Affiliation Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

  • Le Huu Binh, 
  • Thuy-Van T. Duong

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Fig 1

Router nodes placement (RNP) is an important issue in the design and implementation of wireless mesh networks (WMN). This is known as an P-hard problem, which cannot be solved using conventional algorithms. Consequently, approximate optimization strategies are commonly used to solve this problem. With heavy node density and wide-area WMNs, solving the RNP problem using approximation algorithms often faces many difficulties, therefore, a more effective solution is necessary. This motivated us to conduct this work. We propose a new method for solving the RNP problem using reinforcement learning (RL). The RNP problem is modeled as an RL model with environment, agent, action, and reward are equivalent to the network system, routers, coordinate adjustment, and connectivity of the RNP problem, respectively. To the best of our knowledge, this is the first study that applies RL to solve the RNP problem. The experimental results showed that the proposed method increased the network connectivity by up to 22.73% compared to the most recent methods.

Citation: Binh LH, Duong T-VT (2024) A novel and effective method for solving the router nodes placement in wireless mesh networks using reinforcement learning. PLoS ONE 19(4): e0301073. https://doi.org/10.1371/journal.pone.0301073

Editor: Mohammed Balfaqih, University of Jeddah, SAUDI ARABIA

Received: August 9, 2023; Accepted: March 9, 2024; Published: April 10, 2024

Copyright: © 2024 Binh, Duong. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting information files.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Wireless communication is growing and being widely applied in many fields. In the local area network of agencies, businesses, schools, and so on, wireless mesh networks (WMN) [ 1 , 2 ] are the best choice today because of their significant advantages compared to wireless networks using traditional access points. The most notable benefit of the WMN is that it reduces congestion owing to its ability to balance the loads. In addition, the installation of a WMN is very convenient because there is no need to construct wired connections from the gateway to all routers. Fig 1 illustrates an example of a WMN consisting of six mesh routers (represented by r 1 to r 6 ) and eleven mesh clients (represented by c 1 to c 11 ). In addition, at least one the router of the Internet service provider serves as a gateway for clients to access the Internet. If two mesh routers are within range of each other, a wireless link is established between them. A mesh topology consists of of all the mesh routers and wireless links. For a WMN to deliver Internet services, several mesh routers must be connected to the gateway router via wireless or cable links. As shown in Fig 1 , the mesh routers r 1 and r 2 are connected to the gateway router (GPON or FTTh router) via wireless links. Mesh clients are terminal devices that are users of network services. When a mesh client enters the network region, it can be covered by one or more mesh routers; the mesh client connects to the nearest mesh router to access network services.

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https://doi.org/10.1371/journal.pone.0301073.g001

With the rapid development of wireless and mobile communication technologies, network services are becoming more diverse and rich, especially those on fifth-generation (5G) and sixth-generation (6G) wireless network platforms. To effectively provide these services, WMNs must be designed and installed in the most efficient manner possible, allowing network resources to be fully utilized. This is the motivation for researchers to focus on WMN. Some of the most prevalent subjects that have been implemented include network topology control [ 3 – 7 ], router node placement (RNP) [ 8 – 24 ], optimum routing protocols [ 25 – 29 ], and access point allocation [ 30 – 33 ], with the RNP challenge being the most fascinating. Because the RNP problem is known to be NP-hard, it cannot be solved using conventional algorithms. Recently, approximate optimization methods have become useful for solving this problem [ 8 – 12 ]. The authors of [ 8 ] have used the coyote optimization algorithm (COA) to solve the RNP problem. Their proposed method optimizes both network connectivity and user coverage, which are two critical performance criteria. Using MATLAB simulations, the authors demonstrated that the COA algorithm outperformed other well-known optimization algorithms. In [ 10 ], the authors suggested an optimal method called the Chemical Reaction Optimization (CRO) algorithm to solve this problem. The CRO algorithm was inspired by how molecules interact to achieve a low, stable energy state in chemical reactions. In terms of client coverage and network connection, the simulation findings reveal that their suggested approach outperforms the Genetic approach (GA) and Simulated Annealing (SA). Another study employed a genetic algorithm and simulated annealing to discover a low-cost WMN configuration while satisfying restrictions and identifying the number of gateways needed [ 34 ]. Experiments showed that the evolutionary algorithm and simulated annealing were successful in lowering WMN network expenses while maintaining QoS. The new models significantly outperformed the conventional solutions. QoS was also considered in the RNP problem in [ 23 ]. The authors described a unique particle swarm optimization method for improving network connectivity and client coverage. The QoS restrictions for this study are the delay, relay load, and Internet gateway capacity. In [ 35 ], the authors suggested an improved version of the Moth Flame Optimization (MFO) algorithm, namely, Enhanced Chaotic Lévy Opposition-based MFO (ECLO-MFO), for solving the RNP problem. To improve the optimization performance of MFO, the proposed method integrates three strategies: the chaotic map concept, Lévy flying strategy, and Opposition-Based Learning (OBL) technique. The simulation results showed that the proposed algorithm was more efficient than the method of applying popular optimization algorithms.

Based on the results of published works, we find that the method of using approximate optimal algorithms provide good solutions. However, because randomness is used in several steps of the algorithm, the results often differ for different executions. For accurate results, each script must be executed multiple times, and then the average of all executions is obtained. For example, the authors of [ 8 , 11 ] executed each simulation scenario 50 times. Furthermore, with heavy node density and wide-area WMNs, solving the RNP problem with approximation algorithms often presents many difficulties, necessitating a more effective solution. In this paper, we propose a new and effective algorithm to solve this problem. The main contributions of this study are summarized as follows:

  • (i) We proposed a novel and effective method for solving the RNP problem using RL. The RNP problem is modeled as an RL model, with the environment, agent, action, and reward representing the network system, routers, coordinate adjustment, and connectivity respectively, of the RNP problem. To the best of our knowledge, this is the first study to apply reinforcement learning to the RNP problem.
  • (ii) We compared and evaluated the performance of the RNP problem solving method using the heuristic algorithms and the RL method.

The remainder of this paper is organized as follows. The next section describes the formulation of the RNP problem in the WMN. The following sections present our proposed solution and experimental results. Finally, concluding remarks and promising future studies are presented in the last section.

RNP problem

In this section, we formulate the RNP problem in a WMN. First, graph theory was used to describe the WMN. We then define some metrics to use for the objective function of the RNP problem, similar to [ 11 ]. Finally, the RNP problem was formulated as a nonlinear programming problem. For convenience, we define the mathematical symbols shown in Table 1 .

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https://doi.org/10.1371/journal.pone.0301073.t001

Mathematical model of a WMN using graph theory

Consider a WMN comprising m mesh routers, n mesh clients, and k gateway routers. Mathematically, this WMN can be represented as an undirected graph, denoted by G = ( V , E ), where V and E are the vertex and edge sets, respectively. V is equivalent to the set of all nodes in the WMN and is determined by V = R ∪ C ∪ W , where R , C and W are the sets of mesh routers, mesh clients, and gateway routers, respectively. E is equivalent to the set of all wireless links in the WMN and consists of three types: links between mesh routers, links between mesh client and mesh router, and links between gateway and mesh router.

RNP problem formulation

In this section, we formulate the RNP problem using some concepts and metrics from [ 11 ], including the connected router, connected client, connected router ratio, and connected client ratio.

Connected router.

The mesh router r i is a connected router if and only if at least one path exists between it and the gateway router. If we return to the WMN example in Fig 1 , we can see that mesh routers r 1 , r 2 , r 3 , r 4 and r 6 are the connected routers but r 5 is not because no path exists from this mesh router to the gateway router.

Connected router ratio (CRR).

algorithm method of problem solving

Connected client.

algorithm method of problem solving

Connected client ratio (CCR).

algorithm method of problem solving

Formulate the RNP into a nonlinear programming problem.

The RNP problem in the WMN is stated as follows: Consider a case where it is necessary to design and install a WMN with the following assumptions:

  • The network system is located in an area of W × H meters.

algorithm method of problem solving

  • The number of mesh routers was m , and the coverage radius of each mesh router was d r .

algorithm method of problem solving

RL-based mesh router nodes placement

Fundamentals of rl.

algorithm method of problem solving

https://doi.org/10.1371/journal.pone.0301073.g002

RL has been successfully applied to control protocols in wireless networks, typically routing in WMN [ 25 , 27 , 29 ], topology control in wireless sensor networks [ 37 ], improving the performance of energy-harvesting wireless body area networks [ 38 , 39 ]. In this paper, we apply RL to solve the RNP in WMN. Details of this new proposal are presented in the following sections.

Solving the RNP in WMN using RL

The RL has recently been successfully employed to solve technical challenges in wireless communication such as routing [ 27 , 36 ], topology management [ 37 ], and resource allocation. In this study, we use RL to solve the RNP problem. To the best of our knowledge, this is the first study to use RL to address the RNP problem. To do this, the RNP problem must be modeled as a reinforcement learning model with five characteristic factors: agent, environment, state, action, and reward.

An agent is a mesh router that regularly adjusts its coordinates to obtain an optimal topology.

Environment.

In a RL model, the environment is everything that exists around the agent, and it is where the agent acts and interacts. The environment for the RNP problem using RL is the network system, which includes a set of mesh routers, clients, gateway routers, and network area.

Each state is determined by a triple { P c , P r , P w }, where P c , P r and P w are the sets of coordinates for the mesh clients, mesh router, and gateway routers, respectively. The sets are listed in Table 1 .

Action is the way in which the agent interacts with the environment to change its state. For the RNP problem using reinforcement learning, the agents are the mesh routers. Each action was defined by a mesh router that adjusted its coordinates. The set of actions at a specific state s t for each mesh router r i is defined as A t = { mn1s, me1s, ms1s, mw1s, mn2s, me2s, ms2s, mw2s }, where the actions are described in Table 2 , step is a given distance.

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https://doi.org/10.1371/journal.pone.0301073.t002

algorithm method of problem solving

RL algorithm for solving RNP problem.

algorithm method of problem solving

Algorithm 1 The pseudo-code of the reinforcement learning algorithm for solving RNP problem

  • Network area ( W × H );

algorithm method of problem solving

  • The set of mesh routers ( R = { r i | i = 1.. m }), and the coverage radius of each mesh router ( d r );

algorithm method of problem solving

2: while ( learn ≤ numLearn ) do

3:  Randomly choose mesh router r i ∈ R ;

4:   for ( each action a j ∈ A ) do

5:   Update Q ( r i , s t , a t , j ) using ( 11 );

6:   end for

7:  Choose the action a t , k ∈ A t using policy derived from Q-values (e.g., ε -greedy ) according to ( 12 );

8:  Take action a t , k , observe reward R ( r i , s t , a t , k ) and next state s t +1 ;

9:  Update next state ( s t +1 ) to current state ( s t );

10:   learn ← learn + 1;

11: end while

12: P r ← P r in state s t ;

Analyze the computational complexity.

The computational complexity of Algorithm 1 depends mainly on the iteration in Step (2), the number of possible actions in Step (4), and the algorithm for updating the Q value in Step (5). Q ( r i , s t , a t , j ) is updated using Eq. ( 11 ), where the greatest complexity is the calculation of RW ( r i , s t , a t ) according to ( 10 ). RW ( r i , s t , a t ) contains two metrics, CRR and CCR , which are defined by ( 1 ) and ( 5 ), respectively. To determine CRR , we employed a breadth-first search algorithm on a network of m vertices, which is the number of mesh routers. Therefore, the computational complexity was O ( m 2 ). The CCR is calculated using two nested loops of sizes m and n , where n is the number of mesh clients. Therefore, the complexity was O ( m × n ). Because n is always greater than m in a WMN, the computational complexity of RW ( r i , s t , a t ) is O ( m × n ). Consequently, the computational complexity of Algorithm 1 is O ( I × | A | × m × n ), where I is the number of iterations and | A | is the number of possible actions.

The computational complexity of Algorithm 1 is greater than that of the algorithms solving the RNP problem using GA [ 40 ], PSO [ 24 ], and WOA [ 41 ], which we compare in the following section. However, because its computing complexity is a polynomial function, it can be implemented in practice. Furthermore, because the algorithms for solving the RNP problem are run offline, the polynomial complexity is acceptable.

Simulation results and discussion

Simulation scenarios.

The performance of the proposed method was evaluated through a simulation using Python. Our proposed method is compared with the most recent methods that use approximate optimization algorithms to address the RNP problem, including GA [ 40 ], PSO [ 24 ], WOA [ 41 ], and MVO [ 11 ]. All experiments were run on a 3.6 GHz Core i7 CPU computer. The surveyed network instances (NI) are presented in Table 3 . NI-1 and NI-2 were used to investigate the effect of the number of mesh routers on the network performance, with the number of mesh routers ranging from 20 to 45 covering 150 mesh clients (NI-1) and 350 mesh clients (NI-2). NI-3 and NI-4 ware used to study the effect of client density, varying from 100 to 400. In NI-5 and N-6, the effect of the coverage radius of each mesh router was thoroughly examined. The final two NIs were used to investigate the influence of the network area. The parameters of the simulation scenarios and algorithms are presented in Table 4 , where th parameters of the GA, PSO, WOA, and MVO are set as in [ 11 ].

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https://doi.org/10.1371/journal.pone.0301073.t003

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https://doi.org/10.1371/journal.pone.0301073.t004

Simulation results

Topology evaluation..

First, we evaluate the topology obtained when solving the RNP problem using the GA, PSO, WOA, MVO, and our proposed method, which employs reinforcement learning. The results obtained in Fig 3 clearly show topological differences between the methods. These findings were obtained using NI-2 with 30 mesh routers covering 350 mesh clients in an area of 2000 × 2000 [ m 2 ] and a coverage radius of 200 [ m ] for each mesh router. We can observe that the method using RL provides the most optimal topology compared with the methods using approximate optimization algorithms, GA, WOA, PSO, and MVO. Specifically, for the method using reinforcement learning, there are 334 mesh clients covered by at least one mesh router, corresponding to a rate of 95.43%. These values were 292 (83.43%), 309 (88.29%), 313 (89.43%), and 313 (88.86%) for the WOA, GA, PSO, and MVO algorithms, respectively. In addition, the topology of the reinforcement learning method has a wider coverage area than the other methods, which can increase the percentage of clients covered in the case of denser clients.

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(a) WOA, (b) GA, (c) PSO, (d) MVO, and (e) reinforcement learning.

https://doi.org/10.1371/journal.pone.0301073.g003

Impact of mesh router density.

algorithm method of problem solving

The results obtained in Fig 4 clearly show the difference in network connectivity between the proposed method and the method using approximate optimization algorithms. These findings were obtained using NI-1, in which the number of mesh routers varieed from 20 to 45, covering 150 mesh clients in an area of 2000 × 2000 [ m 2 ] and a coverage radius of 200 [ m ] for each mesh router. We can observe that the NC increases proportionally with the number of mesh routers for all methods. This is evident because as the number of mesh routers increases, the coverage area expands, increasing the probability of mesh clients being covered. Comparing the methods of solving RNP problems, the method using RL (legend namely RL-based RNP) gives the highest NC. For example, considering the case of 35 mesh routers, The NC values of the methods using the WOA, PSO, GA, MVO, and RL are 85.64, 87.42, 90.67, 93.42, and 95.68%, respectively. Thus, compared with the method using algorithms WOA, PSO, GA, and MVO, the proposed method improved NC by 10.03, 8.25, 5.01%, and 2.25%, respectively. This is a significant result in improving WMN performance.

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https://doi.org/10.1371/journal.pone.0301073.g004

The results obtained were quite similar for the implementation on NI-2, as shown in Fig 5 . The assumptions of this simulation scenario are the same as those in NI-1, except that the number of mesh clients increases to 350. We can see that the proposed method is highly effective in terms of NC. We can observe that the proposed solution provides high efficiency in terms of NC for most values of the number of mesh routers. The NC of the method using RL increases by an average of 4 to 20% compared with the cases where approximate optimization algorithms are used. As is the case with 35 mesh routers, the NC of the RL is 98.71%. These values of the WOA, PSO, GA, and MVO algorithms were 81.66%, 86.89%, 88.59%, and 94.44% respectively. Thus, the method using RL improved the NC from 4.26% to 17.04%.

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https://doi.org/10.1371/journal.pone.0301073.g005

Based on the findings in Figs 4 and 5 , we can conclude that changing the number of mesh routers affects on network performance in terms of NC. The larger the number of routers, the higher the NC for all investigated RNP problem solving methods. In particular, the method based on RL is the most efficient.

Impact of mesh client density.

In this section, we investigate the effect of client density on network performance. In a WMN, the denser the clients, the greater is the number of connection requests to the routers. As a result, network performance was affected. This is more evident in Fig 6 , where we plot NC as a function of the number of mesh clients. These results are obtained by executing NI-3, where the number of mesh routers is 30, covering 150 to 300 mesh clients. We can easily observe that the method using RL always yields the highest NC regardless of whether the client density is sparse or dense. The NC value of this method from 90.43% to 95.79%. Meanwhile, the NC values for the cases of algorithm WOA, PSO, GA, and MVO are fom 74.59% to 84.08%, from 77.00% to 85.63%, from 82.32% to 91.46%, and from 88.27% to 90.83%, respectively. When 45 mesh routers were used (NI-4), the NC value increased for all methods. This is clearly shown in Fig 7 , where we represent NC versus the number of mesh clients. Comparing the methods, we find that the method using RL outperforms the method using approximate optimal algorithms in terms of NC.

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https://doi.org/10.1371/journal.pone.0301073.g006

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https://doi.org/10.1371/journal.pone.0301073.g007

Impart of the coverage radius of mesh routers.

The coverage radius of the mesh routers is another technological parameter that has a considerable impact on the WMN performance. In this section, we investigate the effect of this technological parameter on the NC metric. The results obtained in Fig 8 clearly show the change in NC with respect to the coverage radius of the mesh routers. These results were implemented using NI-5, which has 30 mesh routers and 150 mesh clients. The coverage radius of each router ranged from 150 to 300 [ m ]. The plots in Fig 8 indicate that the NC increases proportionally to the coverage radius of the mesh routers. This is because expanding the coverage radius increases the likelihood that clients will be covered. As a result, NC increases. In particular, the method using RL yielded the highest NC, reaching close to 100% when the coverage radius was 250 [m] or more. The results are also similar for NI-6, as shown in Fig 9 . The NC value of this NI is greater than that of the NI-5 because this uses more mesh routers. As in the previous scenarios, the method using RL always yields the highest NC.

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https://doi.org/10.1371/journal.pone.0301073.g008

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https://doi.org/10.1371/journal.pone.0301073.g009

Impact of network area.

In the last section, we investigate the effect of network area on the efficiency of RNP problem solving methods. Figs 10 and 11 show the results obtained by executing NI-7 and NI-8, respectively. In these NIs, the network area varies from 2000 × 2000 [ m 2 ] to 3000 × 3000 [ m 2 ]. The NC value decreased according to the network area for all the algorithms. This is because, for a given number of mesh routers, the larger the network area, the lower the percentage of area covered, leading to a decrease in the NC value. However, the NC value of the method using RL is always the largest.

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https://doi.org/10.1371/journal.pone.0301073.g010

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https://doi.org/10.1371/journal.pone.0301073.g011

Based on the above findings, we can conclude that the proposed method, which uses reinforcement learning to solve the RNP problem, is more efficient than a method that uses approximate optimal algorithms. This is a crucial result in the design and implementation of a WMN, which helps find an optimal network topology to exploit network resources more efficiently.

The placement of router nodes in wireless mesh networks is a significant problem that has recently attracted the interest of several research groups. This problem is recognized as NP-hard, and cannot be resolved using conventional algorithms. In this study, we proposed a new and effective method for solving this problem using RL. The process of finding the optimal coordinates for placing mesh routers is modeled as an RL with the main components being environment, agent, action, and reward, which are equivalent to the network system, routers, coordinate adjustment, and network connectivity of the RNP problem, respectively. Simulation results show that our proposed method outperforms the most recent methods in terms of coverage and network connectivity.

In future work, we will continue to develop this method by considering additional constraints on the quality of transmission and load balancing to improve network performance. In addition, the deep reinforcement learning method can also be applied to static and dynamic RNP problems to further improve the performance of the WMN.

Supporting information

S1 dataset..

https://doi.org/10.1371/journal.pone.0301073.s001

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In response to the optimization scheduling problem in the dyeing production process, the author proposes a hierarchical scheduling method for dyeing vats based on genetic algorithm and multi-agent. In this method, a hierarchical scheduling algorithm is used to decompose production scheduling into static and dynamic strategies. The static strategy adopts a genetic algorithm that supports batch processing of multiple products, non equality of equipment, order delivery time, switching cost, and other constraints: Dynamic strategy is a coordinated dynamic optimization algorithm that uses multi-agent systems to support the running status of dye tanks based on static strategies. By solving the algorithm with multiple constraints and dynamic factors in the production process, the final result of the dyeing tank operation task is obtained. The simulation compared pure genetic algorithm with manual scheduling, and the results showed that the hierarchical dynamic scheduling strategy based on data-driven achieved the goal of optimizing the production scheduling of dyeing vats. The practical application results also demonstrate the feasibility of this method.

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  28. Dynamic Scheduling of Multi-agent Electromechanical Production Lines

    In response to the optimization scheduling problem in the dyeing production process, the author proposes a hierarchical scheduling method for dyeing vats based on genetic algorithm and multi-agent. In this method, a hierarchical scheduling algorithm is used to decompose production scheduling into static and dynamic strategies. The static strategy adopts a genetic algorithm that supports batch ...