Using calculator in elementary math teaching
This article discusses whether or not a calculator should be used in teaching math in elementary grades and how to use it wisely.
The "battle" over calculator use
Some people say a calculator enables children to concentrate on understanding and the mathematical concepts instead of spending time on tedious calculations. They say a calculator helps develop number sense, and makes students more confident about their math abilities.
Others are against using calculator in lower level math teaching, saying that it makes children not to learn their basic facts, prevents students from discovering and understanding underlying mathematical concepts and instead encourages them to randomly try different operations without understanding what they"re doing.
They say calculators keep students from benefiting from one of the most important reasons for learning math: to train and discipline the mind and to promote logical reasoning.
There IS a balance
In my opinion, a calculator can be used in the teaching in a good way or a bad way - it all depends on the teacher's approach. The calculator in itself is not bad nor good - it is just a tool. It is used a lot in today's society, so students should learn to use it by the time they finish school.
At the same time, children SHOULD learn their basic facts, be able to do mental calculations, and master long division and other basic paper-pencil algorithms. Mathematics is a field of study that builds on previously established facts. A child that does not know basic multiplication (and division) facts will have a hard time learning factoring, primes, fraction simplification and other fraction operations, the distributive property, etc. etc. Basic algorithms of arithmetic are a needful basis for understanding the corresponding operations with polynomials in algebra. Mastering long precede divisions understanding how fractions correspond to the repeating (non-terminating) decimals, which then paves the way to understanding irrational numbers and real numbers. It all connects together!
For this reason, it is advisable to restrict the calculator use in the lower grades, until children know their basic facts and can add, subtract, multiply, and divide even large numbers with pencil & paper. THIS, in my opinion, builds number sense, as do mental calculations.
This does not mean that you couldn't use calculator occasionally in the elementary grades for special projects, when teaching specific concepts, or for some fun. It could be used for example in science or geography projects, for exploring certain new concepts, for some number games, or checking homework. See below for some ideas.
The discussion here does not apply to graphical calculators in high school. I am strongly in favor of using graphical calculators or a graphing software when studying graphing and calculus. Even there though, one certainly needs to learn the basic idea of how the graphing is done on paper.
Things to keep in mind when using a calculator
When calculator is used more freely, one should pay attention to the following points:
- The calculator is a tool to do calculations. So are the human mind and paper & pencil. Children should be taught when to use a calculator and when mental computing (or even paper & pencil) are more effective or appropriate. Choosing the right "tool" is part of an effective problem-solving process.
- It is very important that students learn how to estimate the result before doing the calculation. It is SO easy to make mistakes when punching the numbers into a calculator. A student must not learn to rely on the calculator without checking that the answer is reasonable.
- A calculator should not be used to try out randomly all possible operations and to check which one produces the right answer. It is crucial that students learn and understand the different mathematical operations so they know WHEN to use which one — and this is true whether the actual calculation is done mentally, on paper, or with a calculator.
Ideas for calculator use in elementary math
If you use these ideas, make sure the children don"t get the idea that a calculator takes away the need to learn mental math. It can serve as a tool to let children explore and observe, but afterwards the teacher should explain concepts, justify the rules of math, and put it all together.
- Kindergartners and first graders can explore numbers by adding 1 repeatedly(which can be done with first pushing 1 + 1 = and then pressing the = button repeatedly) or subtracting 1 repeatedly. Observe their faces when they hit negative numbers! Or, let them investigate what happens to a number when you add zero to it.
- Calculator pattern puzzles: This is an extension of the idea above, where first to third grade children add or subtract the same number repeatedly using a calculator. Children will observe patterns that emerge when you add, say, 2, 5, 10, or 100 repeatedly. For example, they can start at 17 and add 10 repeatedly or start at 149 and subtract 10 repeatedly. Another idea is to let children make their own "pattern puzzles", which are number sequences with a pattern where some numbers are omitted, for example 7, 14, __, __, 35, __, 49. The activity can connect with the idea of multiplication very easily.
- Place value activity with a calculator : Students build numbers with the calculator, for example:
Make a three-digit number with a 6 in the tens place; OR Make a four-digit number larger than 3,500 with a four in the ones place; OR Make a four-digit number with a 3 in the tens and a 9 in the hundreds of places; etc.
Afterwards the teacher lists several numbers on the board and discusses what the numbers that students made in common, such as: all of the numbers are sixty-something. - Write the number one million on the board. Ask students to pick a number that they will add repeatedly with the calculator to reach one million within a reasonable class time. If they pick small numbers, such as 68 or 125, they won't reach it! This can teach children how vast the number one million is.
- When introducing pi, have students measure the circumference and the diameter of several circular objects, and calculate their ratio with a calculator (which saves time and can help keep the focus on the concept).
The Use of Calculators Gets at the Heart of Good Teaching - an article by Susan Ray; no longer online
Comments
I teach in a very small school and I currently teach Algebra 1, 8th grade science, and then Physics to the seniors and I have a small group that has completed high school calculus and we"re doing some Linear Algebra. I, myself, have a Masters in Physics.Before I read some of these posts, I felt that I was pretty rabid anti-calculator, but now I think I"m more middle of the road.
The comments about doing square roots on paper is a good one. No, we don"t need to know how to do that anymore with good precision. However, I would really like all of my students to be able to tell you what two numbers it"s between. Example: 8
Just last year I discovered how to input data in a TI-83 and have it spit out the mean and the standard deviation. In the context of a Physics class, I don"t want to spend a lot of time on things that they should learn in a Statistics class. But if the calculator does it easily, then I can gently introduce the concept and hope that the initial exposure has prepped them for what they need to learn in Stats.In Algebra 1, however, I don"t allow students to use calculators at all. And, it my school, I find that most kids come to my course without a calculator or an inclination to use it. I feel that the basic rundown on the math in Algebra 1 should be: 80% of the numbers should utilize the basic information on a 12x12 multiplication table that kids should have memorized. 15% of the numbers should push beyond those limits. ). And the last 5% should be things that they need a calculator for.
In my opinion, you learn things about numbers when you have to do them in your head. If you want to do the prime factors of 357, you can start with the idea that it is less than 400, so you only have to check up to 20. You also know that it"s odd, so you don"t have to check 2 or any of the events. Then you can realize that you don"t have to check any of the non-prime numbers between 1 and 20. So, you only have to check 3, 5, 7, 11, 13, 17.
This helps students start to develop some fundamental concepts related to sets. There are groups of numbers that share common properties, like evens and odds and primes. This is a deep concept that you might not get if you don"t have to simplify a process for yourself.
But, also, simplifying a process for yourself is really important. Suppose you are head mechanic on a Sprint Cup NASCAR car. They break all of the time. What do you need to do to fix them? What is extraneous to the problem? What is the smallest number of things that you need to test/fix, and in what order should you try them? That"s a long extension from developing algorithmic thought in high school math class. But I would argue that it"s harder to get there if you have been fed answers by a machine your entire life.
I know this is running long. Two more points... I would never use a graphing calculator to actually graph. I have $100 software on my laptop that blows any hand-held graphing calculator out of the water.
Finally, the comment on store clerks and calculators caught my attention. The world certainly needs people to run the cash registers in department stores. But somehow I feel that the goal of getting a good education is so that you can later choose a career that you are passionate about. Cashiers who are passionate about retail are few and far between. I would hope that my students would have a wider set of choices when they finish school.
David Iverson
I think both should be used. I agree we need to learn the basics in elementary school, addition, subtraction, etc.) However, When you go to Macy's, Olive Garden or Mc Donald's, the cashier doesn't use paper and pencil. Computers (calculators ) are used. We live in a computer age. We are no longer in the Industrial Revolution, so let's come into the 21st century.
Hi I"m Kelly. I"m a freshmen in college at St. Charles community college in Missouri. Your site is wonderful. I was looking it over for my younger sister. Something I would really like to tell everyone and anyone who plans on going to college is to stop using a calculator immediately. Only use it for graphing logs and necessary things like that. I finished high school in a calculus class using a calculator for even the simplest multiplication and division problems, and when I got to college I had to start all over in BEGINNING ALGEBRA because I didn't know how to multiply and divide without a calculator. So please do everyone a favor and ask them or tell them to stop using a calculator. They will thank me for it later.
Hello my name is Rafeek and I am a freshman at Hobart and William Smith colleges in Geneva, NY. I am doing a paper on technology and its effects, so I decided to pick the calculator. I came across this site in my research. I want to stress what Kelly said. The same thing happened to me, I was great in high school math, practically aced all math exams, then I came here for orientation and they told me I have to take a math placement test W/OUT a calc. I didn"t realize I couldn"t do a lot of the simple problems because I always plugged it into my calc and got the answer. This is becoming something serious, I already took away my younger brother and sisters calc. and told them until they are in college they will not be using a calc (at least not in front of me). Now I am taking pre-calc. and my goal it is to not use a calc. DO NOT DEPEND ON YOUR CALCULATOR!!!
When at University taking math courses for my BMath we weren't allowed calculators for many of the exams (to prevent people smuggling in pocket computing devices). For anyone doing higher level math I would say that being able to do sums on paper is essential .Emily Bell
I"ve never been good at math and so when i got a hold of my calculator and how encouraging it is in highschool i fell in love with it. that is until i took my college placement test. I did horrible. I couldn"t even remember how to do a simple division problem mentally. The problem with schools today is that they worry and encourage too much about calculators. Students should have a good sturdy base of mental math before they learn to use the calculator and if u ask me K-3 grade isn't enough. it should not be permitted until college.
I am a recent college graduate. My major was Electrical Engineering. As my course of study involved a great deal of mathematics, I feel obligated to speak on this important issue. In my opinion, calculators should never be used for any mathematics class, even at the college level. Using a calculator for any subject will cause the user to become mentally lazy and incapable of basic mathematics skills. You should never use a calculator when learning how to multiply, do long division, or even graph a function."Some people say calculator enables children to concentrate on understanding and studying mathematical concepts instead of spending time on tedious calculations. They say calculator helps develop number sense, and makes students more confident about their math abilities."
The above statement is the total hogwash. The only way to develop number sense and understand mathematical concepts is to pour over hours of tedious calculations. The only way to develop confidence in one's math abilities is to use a pencil and paper whenever you are confronted by a math problem. If a mathematics teacher agrees with the above statement, he or she should be fired immediately. The NCTM should be publicly disgraced for going along with such ruinous ideals.
The only time calculators should be used in school is in the laboratory class when you are doing calculations on numbers with more than 4 significant digits. Otherwise, the student should rely on a paper, a pencil, and his or her brain.
The calculator has no place; NO PLACE; in an elementary school classroom. Period. I am a high school math teacher and the majority of my students have absolutely zero number sense. They"re using calculators to do single-digit multiplication problems they should have rightly memorized in the third grade. They"re helpless without them. I place 100% of the blame on calculator use in the early grades.
My children are 4 and 2. My daughter is going into kindergarten next year, and I"m going to instruct her teachers each year, and periodically throughout the year, she is FORBIDDEN to use a calculator for ANY of her work until she is in high school. There is NOTHING in the elementary or middle school curriculum that requires the use of a calculator.
AS to this statement "National Council of Teachers of Mathematics (1989) has recommended that long division and "practicing tedious pencil-and-paper computations" receive decreased attention in schools, and that calculators be available to all students at all times." My understanding is that this was a reaction to a survey of the time spent on math topics in the classroom and the nearly a third of fourth and fifth grade was spent learning to do division with decimal and double digit divisors (ie 340/.15 or 500/15) Yes teachers were spending more than two months of each of these! This just did not reflect the situation of math in the current world.
Personally, I have seen many great uses for calculators. They allow for error free repetition so that I could discover patterns. Many of the conversions and quick tricks I can do were because I only had a basic calculator all the way through precalculus. BTW, NCMT has also updated its standards to include fluency for math facts in second and fourth grades. As a math tutor I was hearing from parents all the time that children were not spending any time in school memorizing the basic fact.
I would probably have liked it in the long run if I wasn't allowed to use a calculator until at least high school (Geometry for me). You know those Nintendo DS Brainage games? Well they made me realize how awful I am with simple math. I can do it, just takes me a lot longer. Also, I can hardly ever do long division.
As a junior high and high school teacher of Math, Pre-Algebra and Algebra I, I find myself fighting this battle yearly. While yes, calculators offer a quick way of finding answers, I don't know of any problem in any of the three textbooks that I currently use that requires the student to solve long division problems to the upteenth place behind the decimal (which is a common argument).
However I do expect my students to be able to do basic math functions without the use of a calculator. As they get into Algebra, they spend too much time trying to figure out how to do things on the calculator that aren't possible with the calculators they have. I also expect them to show their work on tests and quizzes (so does the new state tests for partial points) so that I KNOW that they know the process. "I used a calculator" does not demonstrate to me that they know process and rules or the "why" it works. to the "look what I found out" and the "ah-ha"s" of mathematics.
I frequently remind students that calculators were invented long after mathematical rules began; therefore, all mathematics can be done without the use of a calculator. Great minds, don't become great by taking the easy way out.
In regards to retail workers, while many customers standing in line would get impatient with the salesperson figuring everything by hand, as a teacher when I go to a food establishment, and that unlucky student of mine is the waiter/waitress/etc. I do expect them to count change back to me. I am mindful of when I do these "checks" and most managers (you know those who can do math without a calculator) are usually appreciative that their employees know how to count change back.
I had to laugh just a bit at the comment regarding "cashiers at Macy"s, Olive Garden, McDonalds...use calculators, computers." True, but that is no argument for their use. Have you ever been at one of these stores when the "computers are down?" Many cashiers cannot figure totals, make change, etc. without a computer to tell them what to do. Strong, basic math skills are very important and IMHO calculator use should be very limited. how some of our young people would fare in a true disaster/emergency when there may not be power, cell phones, computers, internet capability, etc. As a homeschooling parent one of my goals is for my child to have good basic skills firmly in place so they can function well in any subject without electronic help.
I have a boy going in third grade, and I bought him an extremely simple calculator (just +,-,*,/). He"s pretty good at problem solving, he knows his multiplication tables, can do additions and subtractions with 12 digits on paper, is learning on how to do multiplications on paper etc... and I was actually looking for some meaningful problems to solve with a calculator when I found this emotional debate.
Now, I fully agree that a calculator should not be a substitute for learning to do mental operations, and for learning how to do it on paper. You should be able to do these things on your self, even if it is clumsy.But the point is, society advances. Where it was useful to do correctly and quickly sums of 20 numbers on a small note, and people even paid you for that skill 40 years ago, it isn't the case anymore. Most of us don't learn how to kill a rabbit with bow and arrow - while this was an essential skill for our ancestors living in caves.
When I look at the comments here, it seems that the only problems people faced when not being able to calculate without a calculator was in an artificial setting where this was an expressly tested competence. Rabbit hunting with arrow and bow would also pose a problem if this was not taught, and explicitly tested for one or other exam. I think in "real life" it is now important to be handy with a calculator - although one should of course be able to do without, but maybe not *drilled* at doing it efficiently, correctly and fast without.
BTW, who knows still how to take square roots on paper? Isn't this an important skill? And who knows how to use efficiently a slide rule? Or a logarithm table to do multiplications? All these were techniques that were once very useful, and were important to be mastered quickly and efficiently. Now, they belong more to folklore. I don"t say that knowing how to do an addition on paper is folklore, one should know how to do it, but I wonder what"s the reason to be able to do it quickly and efficiently (and hence spend hours training for it).Can"t one use that time now to do more useful things?
I would say, what"s still a practical skill is *mental* calculation, precise mental calculation, and approximate calculation to get an idea of order of magnitude. Whether doing multiplications of two numbers with 6 or 7 digits is still a very useful skill to train onto, I have my doubts - although, again, one should be able to know how it is done.
Things that get interesting with calculators, are constructions like Pascal's triangle, or Fibonacci's series, or factorials, combinations and things like that, and which are too tedious to do by hand.
Patrick Van Esch
Question: What are the main reasons for not using calculators in forms one to three of secondary schools?I'm not quite sure what forms one to three are, but I guess you are talking about high school.
I personally would not deny calculator use of high schoolers. Children need to learn to use calculator, and to use it wisely - which means they should learn WHEN it"s good to use it and when not. Maybe one would deny calculator use in high school if a student was constantly misusing it, in others words using it for 6 x 7 etc., in which case such a student might need to review lower grades math.
I am a current sixth grader, I know most kids my age prefer using a calculator not for checking there work, but doing a large portion of they"re math with calculators. Calculator should be used only for checking work, recently my math teach has practically been forcing us to use TI30 xa calculators,as you know,the school provides a calculator that can add,subtract,multiply,and divide, and that seems to be enough. Lately I have been catching myself relying on calculators to do all mine. work, but today during my math class I decided no more calculator,one problem I had to solve was 3.8892 divided by 3 and I couldn't remember how to do it. And the other day my mom gave me a simple math problem while getting gas and it took me 5 minutes to do this basic addition problem. My parents didn"t use calculators when they were in school and if they didn"t need them then we don"t either. But once all of our current middle schoolers are full grown adults, our school system will see that the adults will be way behind in math while relying on computers, and calculators to do all there deeds. I am officially Anti calculator!
I was lucky enough to learn basic math facts (multiplication, division, fractions, estimation, etc) before getting a calculator in 8th grade, but I grew really dependent on my TI 83 graphing utility for my high school algebra/precalc classes. I would graph the function to find the zeros instead of using the quadratic formula and stuff like that.My freshman calculus class didn't allow calculators, and I failed it. This was after doing quite well in honors high school precalculus. I went into an easier life/social science series (still had to struggle for B"s/C"s when I"d had easy A"s in high school) and eventually repeated the harder calculus class much more prepared. My life/social science series classes allowed 4-function but not graphing utilities Also, in college I had to show my work. to get any credit, even if the answer was right. I think one problem is that I got too hung up on finding the answers rather than learning the process.
My sister on the other hand has had a calculator since 3rd grade, and she literally can"t multiply 6*7 without a calculator or do a word problem, though she does get B"s in high school math.
As a Senior majoring in Early Childhood/Elementary Education, I understand the importance of having the knowledge on how to use a calculator, because yes, we live in an age where technology is widely used. However, like many of you, when I first came to college and had to take exams without use of the calculator, I was in big trouble! I still did very well, but it took me a long time to relearn all of the basic functions of math. From my own personal experiences in the field and through my own courses, I recommend a consistent balance between the two methods!!
I teach mathematics in a college where a calculator is forbidden. Unfortunately many students have been ruined by using a calculator. They have trouble doing even the simplest algebra. This has caused an increase of remedial math in colleges everywhere by up to 95%. There is a book out called "The Deliberate Dumbing Down Of America" written by a former whistle blower from the Department O Education (also known as the DOE which should stand for Dopes Of Education)
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Catalog Information
Title
Elementary Linear Algebra.
(Credit Hours:Lecture Hours:Lab Hours)
Offered
Prerequisite
Minimal learning outcomes
Upon completion of this course, the successful student will be able to:
- Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.
- Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.
- Use Cramer's rule to solve a linear system.
- Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.
- Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.
- Determine whether a given subset of a vector space is a subspace.
- Determine whether a given set of vectors is linearly independent, spans, or is a basis.
- Determine the dimension of a given vector space or of a given subspace.
- Find bases for the null space, row space, and column space of a given matrix, and determine its rank.
- Demonstrate understanding of the Rank-Nullity Theorem and its applications.
- Given a description of a linear transformation, find its matrix representation relative to given bases.
- Demonstrate understanding of the relationship between similarity and change of basis.
- Find the norm of a vector and the angle between two vectors in an inner product space.
- Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.
- Find the orthogonal complement of a given subspace.
- Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.
- Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.
- Determine whether a vector space with a (sesquilinear) form is an inner product space.
- Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in R n and in function spaces that are inner product spaces.
- Use least squares to fit a line ( y = ax + b) to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.
- Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.
- Find (real and complex) eigenvalues and eigenvectors of 2 × 2 or 3 × 3 matrices.
- Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.
- Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.
- Identify symmetric matrices and orthogonal matrices.
- Find a matrix that orthogonally diagonalizes a given symmetric matrix.
- Know and be able to apply the spectral theorem for symmetric matrices.
- Know and be able to apply the Singular Value Decomposition.
- Correctly define terms and give examples relating to the above concepts.
- Prove basic theorems about the above concepts.
- Prove or disprove statements relating to the above concepts.
- Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.
In the traveling salesman problem, to form an optimal route around n cities, you need to choose the best one from (n-1)! options based on time, cost or route length. This problem involves determining a Hamiltonian cycle of minimum length. In such cases, the set of all possible solutions should be represented in the form of a tree - a connected graph that does not contain cycles or loops. The root of the tree unites the entire set of options, and the tops of the tree are subsets of partially ordered solution options.
Purpose of the service. Using the service, you can check your solution or get a new solution to the traveling salesman problem using two methods: the branch and bound method and the Hungarian method.
Mathematical model of the traveling salesman problem
The formulated problem is an integer problem. Let x ij =1 if the traveler moves from the i-th city to the j-th and x ij =0 if this is not the case.Formally, we introduce (n+1) a city located in the same place as the first city, i.e. the distances from (n+1) cities to any other city other than the first are equal to the distances from the first city. Moreover, if you can only leave the first city, then you can only come to the (n+1) city.
Let's introduce additional integer variables equal to the number of visits to this city along the way. u 1 =0, u n +1 =n. In order to avoid closed paths, leave the first city and return to (n+1), we introduce additional restrictions connecting the variables x ij and the variables u i (u i are non-negative integers).
U i -u j +nx ij ≤ n-1, j=2..n+1, i=1..n, i≠j, with i=1 j≠n+1
0≤u i ≤n, x in+1 =x i1 , i=2..n
Methods for solving the traveling salesman problem
- branch and bound method (Little's algorithm or subcycle elimination). An example of a branch and bound solution;
- Hungarian method. An example of a solution using the Hungarian method.
Little's algorithm or subcycle elimination
- Reduction operation along rows: in each row of the matrix, the minimum element d min is found and subtracted from all elements of the corresponding row. Lower limit: H=∑d min .
- Reduction operation by columns: in each column of the matrix, select the minimum element d min and subtract it from all elements of the corresponding column. Lower limit: H=H+∑d min .
- The reduction constant H is the lower bound of the set of all admissible Hamiltonian contours.
- Finding powers of zeros for a matrix given by rows and columns. To do this, temporarily replace the zeros in the matrix with the sign “∞” and find the sum of the minimum elements of the row and column corresponding to this zero.
- Select the arc (i,j) for which the degree of the zero element reaches the maximum value.
- The set of all Hamiltonian contours is divided into two subsets: the subset of Hamiltonian contours containing the arc (i,j) and those not containing it (i*,j*). To obtain a matrix of contours including the arc (i,j), cross out row i and column j in the matrix. To prevent the formation of a non-Hamiltonian contour, replace the symmetric element (j,i) with the sign “∞”. Arc elimination is achieved by replacing the element in the matrix with ∞.
- The matrix of Hamiltonian contours is reduced with a search for the reduction constants H(i,j) and H(i*,j*) .
- The lower bounds of the subset of Hamiltonian contours H(i,j) and H(i*,j*) are compared. If H(i,j)
- If, as a result of branching, a (2x2) matrix is obtained, then the Hamiltonian contour obtained by branching and its length are determined.
- The length of the Hamiltonian contour is compared with the lower boundaries of the dangling branches. If the length of the contour does not exceed their lower boundaries, then the problem is solved. Otherwise, branches of subsets with a lower bound less than the resulting contour are developed until a route with a shorter length is obtained.
Example. Solve the traveling salesman problem with a matrix using Little's algorithm
| 1 | 2 | 3 | 4 | |
| 1 | - | 5 | 8 | 7 |
| 2 | 5 | - | 6 | 15 |
| 3 | 8 | 6 | - | 10 |
| 4 | 7 | 15 | 10 | - |
Solution. Let's take as an arbitrary route: X 0 = (1,2);(2,3);(3,4);(4,5);(5,1). Then F(X 0) = 20 + 14 + 6 + 12 + 5 = 57
To determine the lower bound of the set, we use reduction operation or reducing the matrix row by row, for which it is necessary to find the minimum element in each row of matrix D: d i = min(j) d ij
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 20 | 18 | 12 | 8 | 8 |
| 2 | 5 | M | 14 | 7 | 11 | 5 |
| 3 | 12 | 18 | M | 6 | 11 | 6 |
| 4 | 11 | 17 | 11 | M | 12 | 11 |
| 5 | 5 | 5 | 5 | 5 | M | 5 |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
d j = min(i) d ij
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
| dj | 0 | 0 | 0 | 0 | 0 |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
The elements of the matrix d ij correspond to the distance from point i to point j.
Since there are n cities in the matrix, then D is an nxn matrix with non-negative elements d ij ≥ 0
Each valid route represents a cycle in which the traveling salesman visits the city only once and returns to the original city.
The route length is determined by the expression: F(M k) = ∑d ij
Moreover, each row and column is included in the route only once with the element d ij .
Step #1.
Determining the branching edge
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 12 | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | M | 9 | 2 | 6 | 2 |
| 3 | 6 | 12 | M | 0(5) | 5 | 5 |
| 4 | 0(0) | 6 | 0(0) | M | 1 | 0 |
| 5 | 0(0) | 0(6) | 0(0) | 0(0) | M | 0 |
| dj | 0 | 6 | 0 | 0 | 1 | 0 |
The largest sum of reduction constants is (0 + 6) = 6 for the edge (5,2), therefore, the set is divided into two subsets (5,2) and (5*,2*).
Edge exclusion(5.2) is carried out by replacing the element d 52 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (5*,2*), as a result we obtain a reduced matrix.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 12 | 10 | 4 | 0 | 0 |
| 2 | 0 | M | 9 | 2 | 6 | 0 |
| 3 | 6 | 12 | M | 0 | 5 | 0 |
| 4 | 0 | 6 | 0 | M | 1 | 0 |
| 5 | 0 | M | 0 | 0 | M | 0 |
| dj | 0 | 6 | 0 | 0 | 0 | 6 |
Enabling an edge(5.2) is carried out by eliminating all elements of the 5th row and 2nd column, in which the element d 25 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | 2 | M | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | 0 | M | 1 | 0 |
| dj | 0 | 0 | 0 | 0 | 0 |
The lower bound of the subset (5,2) is equal to: H(5,2) = 35 + 0 = 35 ≤ 41
Since the lower boundary of this subset (5,2) is less than the subset (5*,2*), we include edge (5,2) in the route with a new boundary H = 35
Step #2.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | 9 | 2 | M | 2 |
| 3 | 6 | M | 0(7) | 5 | 5 |
| 4 | 0(0) | 0(9) | M | 1 | 0 |
| dj | 0 | 9 | 2 | 1 | 0 |
The largest sum of reduction constants is (0 + 9) = 9 for the edge (4,3), therefore, the set is divided into two subsets (4,3) and (4*,3*).
Edge exclusion(4.3) is carried out by replacing the element d 43 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (4*,3*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | 2 | M | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | M | M | 1 | 0 |
| dj | 0 | 9 | 0 | 0 | 9 |
Enabling an edge(4.3) is carried out by eliminating all elements of the 4th row and 3rd column, in which the element d 34 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 4 | 5 | d i |
| 1 | M | 4 | 0 | 0 |
| 2 | 0 | 2 | M | 0 |
| 3 | 6 | M | 5 | 5 |
| dj | 0 | 2 | 0 | 7 |
The lower bound of the subset (4,3) is: H(4,3) = 35 + 7 = 42 ≤ 44
Since 42 > 41, we exclude the subset (5,2) for further branching.
We return to the previous plan X 1.
Plan X 1.
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | M | 0 | 0 | M |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 6 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 6 | M | 0 | 5 |
| 4 | 0 | 0 | 0 | M | 1 |
| 5 | 0 | M | 0 | 0 | M |
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 6 | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | M | 9 | 2 | 6 | 2 |
| 3 | 6 | 6 | M | 0(5) | 5 | 5 |
| 4 | 0(0) | 0(6) | 0(0) | M | 1 | 0 |
| 5 | 0(0) | M | 0(0) | 0(0) | M | 0 |
| dj | 0 | 6 | 0 | 0 | 1 | 0 |
The largest sum of reduction constants is (0 + 6) = 6 for the edge (4,2), therefore, the set is divided into two subsets (4,2) and (4*,2*).
Edge exclusion(4.2) is carried out by replacing the element d 42 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (4*,2*), as a result we obtain a reduced matrix.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 6 | 10 | 4 | 0 | 0 |
| 2 | 0 | M | 9 | 2 | 6 | 0 |
| 3 | 6 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | M | 0 | M | 1 | 0 |
| 5 | 0 | M | 0 | 0 | M | 0 |
| dj | 0 | 6 | 0 | 0 | 0 | 6 |
Enabling an edge(4.2) is carried out by eliminating all elements of the 4th row and 2nd column, in which the element d 24 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
The result is another reduced matrix (4 x 4), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | M | 6 | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 5 | 0 | 0 | 0 | M | 0 |
| dj | 0 | 0 | 0 | 0 | 0 |
The lower bound of the subset (4,2) is equal to: H(4,2) = 41 + 0 = 41 ≤ 47
Since the lower boundary of this subset (4,2) is less than the subset (4*,2*), we include edge (4,2) in the route with a new boundary H = 41
Step #2.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0(9) | 4 |
| 2 | 0(6) | 9 | M | 6 | 6 |
| 3 | 6 | M | 0(5) | 5 | 5 |
| 5 | 0(0) | 0(9) | 0(0) | M | 0 |
| dj | 0 | 9 | 0 | 5 | 0 |
The largest sum of reduction constants is (4 + 5) = 9 for the edge (1,5), therefore, the set is divided into two subsets (1,5) and (1*,5*).
Edge exclusion(1.5) is carried out by replacing the element d 15 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (1*,5*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | M | 4 |
| 2 | 0 | 9 | M | 6 | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 5 | 0 | 0 | 0 | M | 0 |
| dj | 0 | 0 | 0 | 5 | 9 |
Enabling an edge(1.5) is carried out by eliminating all elements of the 1st row and 5th column, in which the element d 51 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
As a result, we obtain another reduced matrix (3 x 3), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 3 | 4 | d i |
| 2 | 0 | 9 | M | 0 |
| 3 | 6 | M | 0 | 0 |
| 5 | M | 0 | 0 | 0 |
| dj | 0 | 0 | 0 | 0 |
The lower bound of the subset (1,5) is equal to: H(1,5) = 41 + 0 = 41 ≤ 50
Since the lower boundary of this subset (1,5) is less than the subset (1*,5*), we include edge (1,5) in the route with a new boundary H = 41
Step #3.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | d i |
| 2 | 0(15) | 9 | M | 9 |
| 3 | 6 | M | 0(6) | 6 |
| 5 | M | 0(9) | 0(0) | 0 |
| dj | 6 | 9 | 0 | 0 |
The largest sum of reduction constants is (9 + 6) = 15 for edge (2,1), therefore, the set is divided into two subsets (2,1) and (2*,1*).
Edge exclusion(2.1) is carried out by replacing the element d 21 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (2*,1*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | d i |
| 2 | M | 9 | M | 9 |
| 3 | 6 | M | 0 | 0 |
| 5 | M | 0 | 0 | 0 |
| dj | 6 | 0 | 0 | 15 |
Enabling an edge(2.1) is carried out by eliminating all elements of the 2nd row and 1st column, in which the element d 12 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
As a result, we obtain another reduced matrix (2 x 2), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 3 | 4 | d i |
| 3 | M | 0 | 0 |
| 5 | 0 | 0 | 0 |
| dj | 0 | 0 | 0 |
∑d i + ∑d j = 0
The lower bound of the subset (2,1) is equal to: H(2,1) = 41 + 0 = 41 ≤ 56
Since the lower boundary of this subset (2,1) is less than the subset (2*,1*), we include the edge (2,1) in the route with a new boundary H = 41.
In accordance with this matrix, we include edges (3,4) and (5,3) in the Hamiltonian route.
As a result, along the branching tree of the Hamiltonian cycle, the edges form:
(4,2), (2,1), (1,5), (5,3), (3,4). The route length is F(Mk) = 41
Decision tree.
| 1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (5*,2*), H=41 | (5,2) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (4*,2*), H=47 | (4,2) | (4*,3*), H=44 | (4,3) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (1*,5*), H=50 | (1,5) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (2*,1*), H=56 | (2,1) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (3,4) | (3*,4*), H=41 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (5,3) | (5*,3*), H=41 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The following are also used with this calculator:
Graphical method for solving ZLP
Simplex method for solving ZLP
Solving a matrix game
Using the online service, you can determine the price of a matrix game (lower and upper bounds), check for the presence of a saddle point, find a solution to a mixed strategy using the following methods: minimax, simplex method, graphical (geometric) method, Brown's method.
Extremum of a function of two variables
Dynamic programming problems
The first stage of solving the transport problem is to determine its type (open or closed, or otherwise balanced or unbalanced). Approximate methods ( methods for finding a reference plan) allow for second stage of solution in a small number of steps obtain an acceptable, but not always optimal, solution to the problem. This group of methods includes the following methods:
- deletion (double preference method);
- northwest corner;
- minimum element;
- Vogel approximations.
Reference solution to the transport problem
The reference solution to the transport problem is any feasible solution for which the condition vectors corresponding to the positive coordinates are linearly independent. To check the linear independence of the vectors of conditions corresponding to the coordinates of an admissible solution, cycles are used.Cycle A sequence of cells in a transport task table is called in which two and only adjacent cells are located in the same row or column, and the first and last are also in the same row or column. A system of vectors of transport problem conditions is linearly independent if and only if no cycle can be formed from the corresponding cells of the table. Therefore, an admissible solution to the transport problem, i=1,2,...,m; j=1,2,...,n is a reference only if no cycle can be formed from the table cells occupied by it.
Approximate methods for solving the transport problem.
Cross-out method (double preference method). If there is one occupied cell in a row or column of a table, then it cannot be included in any cycle, since a cycle has two and only two cells in each column. Therefore, you can cross out all the rows of the table that contain one occupied cell, then cross out all the columns that contain one occupied cell, then return to the rows and continue crossing out rows and columns. If, as a result of deleting, all rows and columns are crossed out, it means that from the occupied cells of the table it is impossible to select a part that forms a cycle, and the system of corresponding vectors of conditions is linearly independent, and the solution is a reference one. If, after deleting, some cells remain, then these cells form a cycle, the system of corresponding vectors of conditions is linearly dependent, and the solution is not a reference one.
Northwest Angle Method consists of sequentially enumerating the rows and columns of the transport table, starting from the left column and the top row, and writing out the maximum possible shipments in the corresponding cells of the table so that the capabilities of the supplier or the needs of the consumer stated in the task are not exceeded. In this method, no attention is paid to delivery prices, since further optimization of shipments is assumed.
Minimal Element Method. Featuring simplicity this method still more effective than, for example, the Northwest Angle method. Moreover, the minimum element method is clear and logical. Its essence is that in the transport table, the cells with the lowest tariffs are filled in first, and then the cells with high tariffs. That is, we choose transportation with the minimum cost of cargo delivery. This is an obvious and logical move. True, it does not always lead to the optimal plan.
Vogel approximation method. With the Vogel approximation method, at each iteration, the difference between the two minimum tariffs written in them is found for all columns and all rows. These differences are recorded in a specially designated row and column in the table of problem conditions. Among the indicated differences, the minimum is chosen. In the row (or column) to which this difference corresponds, the minimum tariff is determined. The cell in which it is written is filled in at this iteration.
Example No. 1. Tariff matrix (here the number of suppliers is 4, the number of stores is 6):
| 1 | 2 | 3 | 4 | 5 | 6 | Reserves | |
| 1 | 3 | 20 | 8 | 13 | 4 | 100 | 80 |
| 2 | 4 | 4 | 18 | 14 | 3 | 0 | 60 |
| 3 | 10 | 4 | 18 | 8 | 6 | 0 | 30 |
| 4 | 7 | 19 | 17 | 10 | 1 | 100 | 60 |
| Needs | 10 | 30 | 40 | 50 | 70 | 30 |
∑a = 80 + 60 + 30 + 60 = 230
∑b = 10 + 30 + 40 + 50 + 70 + 30 = 230
The balance condition is met. Supplies equal needs. So, the model of the transport problem is closed. If the model were open, it would be necessary to introduce additional suppliers or consumers.
On second stage The reference plan is searched using the methods given above (the most common is the least cost method).
To demonstrate the algorithm, we present only a few iterations.
Iteration No. 1. The minimum matrix element is zero. For this element, inventories are 60 and requirements are 30. We select the minimum number 30 from them and subtract it (see table). At the same time, we cross out the sixth column from the table (its needs are equal to 0).
| 3 | 20 | 8 | 13 | 4 | x | 80 |
| 4 | 4 | 18 | 14 | 3 | 0 | 60 - 30 = 30 |
| 10 | 4 | 18 | 8 | 6 | x | 30 |
| 7 | 19 | 17 | 0 | 1 | x | 60 |
| 10 | 30 | 40 | 50 | 70 | 30 - 30 = 0 | 0 |
Iteration No. 2. Again we are looking for the minimum (0). From the pair (60;50) we select the minimum number 50. Cross out the fifth column.
| 3 | 20 | 8 | x | 4 | x | 80 |
| 4 | 4 | 18 | x | 3 | 0 | 30 |
| 10 | 4 | 18 | x | 6 | x | 30 |
| 7 | 19 | 17 | 0 | 1 | x | 60 - 50 = 10 |
| 10 | 30 | 40 | 50 - 50 = 0 | 70 | 0 | 0 |
Iteration No. 3. We continue the process until we have selected all the needs and supplies.
Iteration No. N. The element you are looking for is 8. For this element, supplies are equal to requirements (40).
| 3 | x | 8 | x | 4 | x | 40 - 40 = 0 |
| x | x | x | x | 3 | 0 | 0 |
| x | 4 | x | x | x | x | 0 |
| x | x | x | 0 | 1 | x | 0 |
| 0 | 0 | 40 - 40 = 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 3 | 4 | 5 | 6 | Reserves | |
| 1 | 3 | 20 | 8 | 13 | 4 | 100 | 80 |
| 2 | 4 | 4 | 18 | 14 | 3 | 0 | 60 |
| 3 | 10 | 4 | 18 | 8 | 6 | 0 | 30 |
| 4 | 7 | 19 | 17 | 0 | 1 | 100 | 60 |
| Needs | 10 | 30 | 40 | 50 | 70 | 30 |
Let's count the number of occupied cells of the table, there are 8 of them, but it should be m + n - 1 = 9. Therefore, the support plan is degenerate. We are making a new plan. Sometimes you have to build several reference plans before finding a non-degenerate one.
| 1 | 2 | 3 | 4 | 5 | 6 | Reserves | |
| 1 | 3 | 20 | 8 | 13 | 4 | 100 | 80 |
| 2 | 4 | 4 | 18 | 14 | 3 | 0 | 60 |
| 3 | 10 | 4 | 18 | 8 | 6 | 0 | 30 |
| 4 | 7 | 19 | 17 | 0 | 1 | 100 | 60 |
| Needs | 10 | 30 | 40 | 50 | 70 | 30 |
As a result, the first support plan is obtained, which is valid, since the number of occupied cells of the table is 9 and corresponds to the formula m + n - 1 = 6 + 4 - 1 = 9, i.e. the reference plan is non-degenerate.
Third stage consists in improving the found reference plan. Here they use the potential method or distribution method. At this stage, the correctness of the solution can be monitored through the cost function F(x) . If it decreases (subject to minimizing costs), then the solution is correct.
Example No. 2. Using the minimum tariff method, present an initial plan for solving a transportation problem. Check for optimality using the potential method.
| 30 | 50 | 70 | 10 | 30 | 10 | |
| 40 | 2 | 4 | 6 | 1 | 1 | 2 |
| 80 | 3 | 4 | 5 | 9 | 9 | 6 |
| 60 | 4 | 3 | 2 | 7 | 8 | 7 |
| 20 | 5 | 1 | 3 | 5 | 7 | 9 |
Example No. 3. Four confectionery factories can produce three types of confectionery products. The production costs of one quintal (quintal) of confectionery products by each factory, the production capacity of the factories (quintal per month) and the daily requirements for confectionery products (quintal per month) are indicated in the table. Draw up a confectionery production plan that minimizes total production costs.
Note. Here, you can first transpose the cost table, since for the classical formulation of the transport problem, capacities (production) come first, and then consumers.
Example No. 4. For the construction of facilities, bricks are supplied from three (I, II, III) factories. Factories have 50, 100 and 50 thousand units in warehouses, respectively. bricks Objects require 50, 70, 40 and 40 thousand pieces, respectively. bricks Tariffs (den. units/thousand units) are shown in the table. Create a transportation plan that minimizes total transportation costs.
will be closed if:A) a=40, b=45
B) a=45, b=40
B) a=11, b=12
Condition of the closed transport problem: ∑a = ∑b
We find, ∑a = 35+20+b = 55+b; ∑b = 60+a
We get: 55+b = 60+a
Equality will be observed only when a=40, b=45
