Hungarian Method

The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primal-dual alternatives. In 1955, Harold Kuhn used the term “Hungarian method” to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry. Let’s go through the steps of the Hungarian method with the help of a solved example.

Hungarian Method to Solve Assignment Problems

The Hungarian method is a simple way to solve assignment problems. Let us first discuss the assignment problems before moving on to learning the Hungarian method.

What is an Assignment Problem?

A transportation problem is a type of assignment problem. The goal is to allocate an equal amount of resources to the same number of activities. As a result, the overall cost of allocation is minimised or the total profit is maximised.

Because available resources such as workers, machines, and other resources have varying degrees of efficiency for executing different activities, and hence the cost, profit, or loss of conducting such activities varies.

Assume we have ‘n’ jobs to do on ‘m’ machines (i.e., one job to one machine). Our goal is to assign jobs to machines for the least amount of money possible (or maximum profit). Based on the notion that each machine can accomplish each task, but at variable levels of efficiency.

Hungarian Method Steps

Check to see if the number of rows and columns are equal; if they are, the assignment problem is considered to be balanced. Then go to step 1. If it is not balanced, it should be balanced before the algorithm is applied.

Step 1 – In the given cost matrix, subtract the least cost element of each row from all the entries in that row. Make sure that each row has at least one zero.

Step 2 – In the resultant cost matrix produced in step 1, subtract the least cost element in each column from all the components in that column, ensuring that each column contains at least one zero.

Step 3 – Assign zeros

  • Analyse the rows one by one until you find a row with precisely one unmarked zero. Encircle this lonely unmarked zero and assign it a task. All other zeros in the column of this circular zero should be crossed out because they will not be used in any future assignments. Continue in this manner until you’ve gone through all of the rows.
  • Examine the columns one by one until you find one with precisely one unmarked zero. Encircle this single unmarked zero and cross any other zero in its row to make an assignment to it. Continue until you’ve gone through all of the columns.

Step 4 – Perform the Optimal Test

  • The present assignment is optimal if each row and column has exactly one encircled zero.
  • The present assignment is not optimal if at least one row or column is missing an assignment (i.e., if at least one row or column is missing one encircled zero). Continue to step 5. Subtract the least cost element from all the entries in each column of the final cost matrix created in step 1 and ensure that each column has at least one zero.

Step 5 – Draw the least number of straight lines to cover all of the zeros as follows:

(a) Highlight the rows that aren’t assigned.

(b) Label the columns with zeros in marked rows (if they haven’t already been marked).

(c) Highlight the rows that have assignments in indicated columns (if they haven’t previously been marked).

(d) Continue with (b) and (c) until no further marking is needed.

(f) Simply draw the lines through all rows and columns that are not marked. If the number of these lines equals the order of the matrix, then the solution is optimal; otherwise, it is not.

Step 6 – Find the lowest cost factor that is not covered by the straight lines. Subtract this least-cost component from all the uncovered elements and add it to all the elements that are at the intersection of these straight lines, but leave the rest of the elements alone.

Step 7 – Continue with steps 1 – 6 until you’ve found the highest suitable assignment.

Hungarian Method Example

Use the Hungarian method to solve the given assignment problem stated in the table. The entries in the matrix represent each man’s processing time in hours.

\(\begin{array}{l}\begin{bmatrix} & I & II & III & IV & V \\1 & 20 & 15 & 18 & 20 & 25 \\2 & 18 & 20 & 12 & 14 & 15 \\3 & 21 & 23 & 25 & 27 & 25 \\4 & 17 & 18 & 21 & 23 & 20 \\5 & 18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)

With 5 jobs and 5 men, the stated problem is balanced.

\(\begin{array}{l}A = \begin{bmatrix}20 & 15 & 18 & 20 & 25 \\18 & 20 & 12 & 14 & 15 \\21 & 23 & 25 & 27 & 25 \\17 & 18 & 21 & 23 & 20 \\18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)

Subtract the lowest cost element in each row from all of the elements in the given cost matrix’s row. Make sure that each row has at least one zero.

\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 5 & 10 \\6 & 8 & 0 & 2 & 3 \\0 & 2 & 4 & 6 & 4 \\0 & 1 & 4 & 6 & 3 \\2 & 2 & 0 & 3 & 4 \\\end{bmatrix}\end{array} \)

Subtract the least cost element in each Column from all of the components in the given cost matrix’s Column. Check to see if each column has at least one zero.

\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 3 & 7 \\6 & 8 & 0 & 0 & 0 \\0 & 2 & 4 & 4 & 1 \\0 & 1 & 4 & 4 & 0 \\2 & 2 & 0 & 1 & 1 \\\end{bmatrix}\end{array} \)

When the zeros are assigned, we get the following:

Hungarian Method

The present assignment is optimal because each row and column contain precisely one encircled zero.

Where 1 to II, 2 to IV, 3 to I, 4 to V, and 5 to III are the best assignments.

Hence, z = 15 + 14 + 21 + 20 + 16 = 86 hours is the optimal time.

Practice Question on Hungarian Method

Use the Hungarian method to solve the following assignment problem shown in table. The matrix entries represent the time it takes for each job to be processed by each machine in hours.

\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)

Stay tuned to BYJU’S – The Learning App and download the app to explore all Maths-related topics.

Frequently Asked Questions on Hungarian Method

What is hungarian method.

The Hungarian method is defined as a combinatorial optimization technique that solves the assignment problems in polynomial time and foreshadowed subsequent primal–dual approaches.

What are the steps involved in Hungarian method?

The following is a quick overview of the Hungarian method: Step 1: Subtract the row minima. Step 2: Subtract the column minimums. Step 3: Use a limited number of lines to cover all zeros. Step 4: Add some more zeros to the equation.

What is the purpose of the Hungarian method?

When workers are assigned to certain activities based on cost, the Hungarian method is beneficial for identifying minimum costs.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

write the steps of the hungarian method for assignment problems

  • Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

close

  • Practice Mathematical Algorithm
  • Mathematical Algorithms
  • Pythagorean Triplet
  • Fibonacci Number
  • Euclidean Algorithm
  • LCM of Array
  • GCD of Array
  • Binomial Coefficient
  • Catalan Numbers
  • Sieve of Eratosthenes
  • Euler Totient Function
  • Modular Exponentiation
  • Modular Multiplicative Inverse
  • Stein's Algorithm
  • Juggler Sequence
  • Chinese Remainder Theorem
  • Quiz on Fibonacci Numbers

Related Articles

  • Solve Coding Problems
  • Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
  • Maximum number of edges in Bipartite graph
  • Types of Graphs with Examples
  • Count of nodes with maximum connection in an undirected graph
  • Erdos Renyl Model (for generating Random Graphs)
  • Clustering Coefficient in Graph Theory
  • Convert the undirected graph into directed graph such that there is no path of length greater than 1
  • Cost of painting n * m grid
  • Count of Disjoint Groups by grouping points that are at most K distance apart
  • Maximize count of nodes disconnected from all other nodes in a Graph
  • Find node having maximum number of common nodes with a given node K
  • Number of Simple Graph with N Vertices and M Edges
  • Ways to Remove Edges from a Complete Graph to make Odd Edges
  • Program to find the number of region in Planar Graph
  • Maximum positive integer divisible by C and is in the range [A, B]
  • Program to check if N is a Tetradecagonal Number
  • Check if it is possible to make x and y zero at same time with given operation
  • Calculate sum of all integers from 1 to N, excluding perfect power of 2
  • Find the sum of the first N Centered Pentagonal Number

Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)

Given a 2D array , arr of size N*N where arr[i][j] denotes the cost to complete the j th job by the i th worker. Any worker can be assigned to perform any job. The task is to assign the jobs such that exactly one worker can perform exactly one job in such a way that the total cost of the assignment is minimized.

Input: arr[][] = {{3, 5}, {10, 1}} Output: 4 Explanation: The optimal assignment is to assign job 1 to the 1st worker, job 2 to the 2nd worker. Hence, the optimal cost is 3 + 1 = 4. Input: arr[][] = {{2500, 4000, 3500}, {4000, 6000, 3500}, {2000, 4000, 2500}} Output: 4 Explanation: The optimal assignment is to assign job 2 to the 1st worker, job 3 to the 2nd worker and job 1 to the 3rd worker. Hence, the optimal cost is 4000 + 3500 + 2000 = 9500.

Different approaches to solve this problem are discussed in this article .

Approach: The idea is to use the Hungarian Algorithm to solve this problem. The algorithm is as follows:

  • For each row of the matrix, find the smallest element and subtract it from every element in its row.
  • Repeat the step 1 for all columns.
  • Cover all zeros in the matrix using the minimum number of horizontal and vertical lines.
  • Test for Optimality : If the minimum number of covering lines is N , an optimal assignment is possible. Else if lines are lesser than N , an optimal assignment is not found and must proceed to step 5.
  • Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Consider an example to understand the approach:

Let the 2D array be: 2500 4000 3500 4000 6000 3500 2000 4000 2500 Step 1: Subtract minimum of every row. 2500, 3500 and 2000 are subtracted from rows 1, 2 and 3 respectively. 0   1500  1000 500  2500   0 0   2000  500 Step 2: Subtract minimum of every column. 0, 1500 and 0 are subtracted from columns 1, 2 and 3 respectively. 0    0   1000 500  1000   0 0   500  500 Step 3: Cover all zeroes with minimum number of horizontal and vertical lines. Step 4: Since we need 3 lines to cover all zeroes, the optimal assignment is found.   2500   4000  3500  4000  6000   3500   2000  4000  2500 So the optimal cost is 4000 + 3500 + 2000 = 9500

For implementing the above algorithm, the idea is to use the max_cost_assignment() function defined in the dlib library . This function is an implementation of the Hungarian algorithm (also known as the Kuhn-Munkres algorithm) which runs in O(N 3 ) time. It solves the optimal assignment problem. 

Below is the implementation of the above approach:

Time Complexity: O(N 3 ) Auxiliary Space: O(N 2 )

Please Login to comment...

  • Mathematical
  • rahul_chauhan_1998

Improve your Coding Skills with Practice

 alt=

What kind of Experience do you want to share?

HungarianAlgorithm.com

Index     Assignment problem     Hungarian algorithm     Solve online    

The Hungarian algorithm: An example

We consider an example where four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker. The matrix below shows the cost of assigning a certain worker to a certain job. The objective is to minimize the total cost of the assignment.

Below we will explain the Hungarian algorithm using this example. Note that a general description of the algorithm can be found here .

Step 1: Subtract row minima

We start with subtracting the row minimum from each row. The smallest element in the first row is, for example, 69. Therefore, we substract 69 from each element in the first row. The resulting matrix is:

Step 2: Subtract column minima

Similarly, we subtract the column minimum from each column, giving the following matrix:

Step 3: Cover all zeros with a minimum number of lines

We will now determine the minimum number of lines (horizontal or vertical) that are required to cover all zeros in the matrix. All zeros can be covered using 3 lines:

Step 4: Create additional zeros

First, we find that the smallest uncovered number is 6. We subtract this number from all uncovered elements and add it to all elements that are covered twice. This results in the following matrix:

Now we return to Step 3.

Again, We determine the minimum number of lines required to cover all zeros in the matrix. Now there are 4 lines required:

Because the number of lines required (4) equals the size of the matrix ( n =4), an optimal assignment exists among the zeros in the matrix. Therefore, the algorithm stops.

The optimal assignment

The following zeros cover an optimal assignment:

This corresponds to the following optimal assignment in the original cost matrix:

Thus, worker 1 should perform job 3, worker 2 job 2, worker 3 job 1, and worker 4 should perform job 4. The total cost of this optimal assignment is to 69 + 37 + 11 + 23 = 140.

Solve your own problem online

HungarianAlgorithm.com © 2013-2024

Book cover

50 Years of Integer Programming 1958-2008 pp 29–47 Cite as

The Hungarian Method for the Assignment Problem

  • Harold W. Kuhn 9  
  • First Online: 01 January 2009

9324 Accesses

182 Citations

8 Altmetric

This paper has always been one of my favorite “children,” combining as it does elements of the duality of linear programming and combinatorial tools from graph theory. It may be of some interest to tell the story of its origin.

  • Graph Theory
  • Combinatorial Optimization
  • Integer Program
  • Assignment Problem
  • National Bureau

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in via an institution .

Buying options

  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Unable to display preview.  Download preview PDF.

H.W. Kuhn, On the origin of the Hungarian Method , History of mathematical programming; a collection of personal reminiscences (J.K. Lenstra, A.H.G. Rinnooy Kan, and A. Schrijver, eds.), North Holland, Amsterdam, 1991, pp. 77–81.

Google Scholar  

A. Schrijver, Combinatorial optimization: polyhedra and efficiency , Vol. A. Paths, Flows, Matchings, Springer, Berlin, 2003.

MATH   Google Scholar  

Download references

Author information

Authors and affiliations.

Princeton University, Princeton, USA

Harold W. Kuhn

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Harold W. Kuhn .

Editor information

Editors and affiliations.

Inst. Informatik, Universität Köln, Pohligstr. 1, Köln, 50969, Germany

Michael Jünger

Fac. Sciences de Base (FSB), Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

Thomas M. Liebling

Ensimag, Institut Polytechnique de Grenoble, avenue Félix Viallet 46, Grenoble CX 1, 38031, France

Denis Naddef

School of Industrial &, Georgia Institute of Technology, Ferst Drive NW., 765, Atlanta, 30332-0205, USA

George L. Nemhauser

IBM Corporation, Route 100 294, Somers, 10589, USA

William R. Pulleyblank

Inst. Informatik, Universität Heidelberg, Im Neuenheimer Feld 326, Heidelberg, 69120, Germany

Gerhard Reinelt

ed Informatica, CNR - Ist. Analisi dei Sistemi, Viale Manzoni 30, Roma, 00185, Italy

Giovanni Rinaldi

Center for Operations Reserach &, Université Catholique de Louvain, voie du Roman Pays 34, Leuven, 1348, Belgium

Laurence A. Wolsey

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter.

Kuhn, H.W. (2010). The Hungarian Method for the Assignment Problem. In: Jünger, M., et al. 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68279-0_2

Download citation

DOI : https://doi.org/10.1007/978-3-540-68279-0_2

Published : 06 November 2009

Publisher Name : Springer, Berlin, Heidelberg

Print ISBN : 978-3-540-68274-5

Online ISBN : 978-3-540-68279-0

eBook Packages : Mathematics and Statistics Mathematics and Statistics (R0)

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

  • Publish with us

Policies and ethics

  • Find a journal
  • Track your research

Hungarian Method: Assignment Problem

Hungarian Method is an efficient method for solving assignment problems .

This method is based on the following principle:

  • If a constant is added to, or subtracted from, every element of a row and/or a column of the given cost matrix of an assignment problem, the resulting assignment problem has the same optimal solution as the original problem.

Hungarian Algorithm

The objective of this section is to examine a computational method - an algorithm - for deriving solutions to the assignment problems. The following steps summarize the approach:

Steps in Hungarian Method

1. Identify the minimum element in each row and subtract it from every element of that row.

2. Identify the minimum element in each column and subtract it from every element of that column.

3. Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way:

  • For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.
  • If for a row and a column, there are two or more zeros and one cannot be chosen by inspection, then you are at liberty to choose the cell arbitrarily for assignment.

4. An optimal assignment is found, if the number of assigned cells equals the number of rows (and columns). In case you have chosen a zero cell arbitrarily, there may be alternate optimal solutions. If no optimal solution is found, go to step 5.

5. Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix obtained from step 3 by adopting the following procedure:

  • Mark all the rows that do not have assignments.
  • Mark all the columns (not already marked) which have zeros in the marked rows.
  • Mark all the rows (not already marked) that have assignments in marked columns.
  • Repeat steps 5 (i) to (iii) until no more rows or columns can be marked.
  • Draw straight lines through all unmarked rows and marked columns.

You can also draw the minimum number of lines by inspection.

6. Select the smallest element from all the uncovered elements. Subtract this smallest element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment.

7. Go to step 3 and repeat the procedure until you arrive at an optimal assignment.

For the time being we assume that number of jobs is equal to number of machines or persons. Later in the chapter, we will remove this restrictive assumption and consider a special case where no. of facilities and tasks are not equal.

Share This Article

Operations Research Simplified Back Next

Goal programming Linear programming Simplex Method Transportation Problem

Assignment problem: Hungarian method 3

Unmarkierte Änderungen werden auf dieser Seite angezeigt

Assignment problem: Hungarian Method Nui Ruppert (Mtk_Nr.: 373224) David Lenh (Mtk_Nr.: 368343) Amir Farshchi Tabrizi (Mtk-Nr.: 372894)

In this OR-Wiki entry we're going to explain the Hungarian method with 3 examples. In the first example you'll find the optimal solution after a few steps with the help of the reduced matrix. The second example illustrates a complex case where you need to proceed all the steps of the algorithm to get to an optimal solution. Finally in the third example we will show how to solve a maximization problem with the Hungarian method.

Inhaltsverzeichnis

  • 1 Introduction
  • 2 Example 1 – Minimization problem
  • 3 Example 2 – Minimazation problem
  • 4 Example 3 – Maximization problem
  • 6 References

Introduction

The Hungarian method is a combinatorial optimization algorithm which was developed and published by Harold Kuhn in 1955. This method was originally invented for the best assignment of a set of persons to a set of jobs. It is a special case of the transportation problem. The algorithm finds an optimal assignment for a given “n x n” cost matrix. “Assignment problems deal with the question how to assign n items (e.g. jobs) to n machines (or workers) in the best possible way. […] Mathematically an assignment is nothing else than a bijective mapping of a finite set into itself […]” [1]

The assignment constraints are mathematically defined as:

To make clear how to solve an assignment problem with the Hungarian algorithm we will show you the different cases with several examples which can occur .

Example 1 – Minimization problem

In this example we have to assign 4 workers to 4 machines. Each worker causes different costs for the machines. Your goal is to minimize the total cost to the condition that each machine goes to exactly 1 person and each person works at exactly 1 machine. For comprehension: Worker 1 causes a cost of 6 for machine 1 and so on …

To solve the problem we have to perform the following steps:

Step 1 – Subtract the row minimum from each row.

Step 2 – Subtract the column minimum from each column from the reduced matrix.

The idea behind these 2 steps is to simplify the matrix since the solution of the reduced matrix will be exactly the same as of the original matrix.

Step 3 – Assign one “0” to each row & column.

Now that we have simplified the matrix we can assign each worker with the minimal cost to each machine which is represented by a “0”.

- In the first row we have one assignable “0” therefore we assign it to worker 3 .

- In the second row we also only have one assignable “0” therefore we assign it to worker 4 .

- In the third row we have two assignable “0”. We leave it as it is for now.

- In the fourth row we have one assignable “0” therefore we assign it. Consider that we can only assign each worker to each machine hence we can’t allocate any other “0” in the first column.

- Now we go back to the third row which now only has one assignable “0” for worker 2 .

As soon as we can assign each worker to one machine, we have the optimal solution . In this case there is no need to proceed any further steps. Remember also, if we decide on an arbitrary order in which we start allocating the “0”s then we may get into a situation where we have 3 assignments as against the possible 4. If we assign a “0” in the third row to worker 1 we wouldn’t be able to allocate any “0”s in column one and row two.

The rule to assign the “0”:

- If there is an assignable “0”, only 1 assignable “0” in any row or any column, assign it.

- If there are more than 1, leave it and proceed.

This rule would try to give us as many assignments as possible.

Now there are also cases where you won’t get an optimal solution for a reduced matrix after one iteration. The following example will explain it.

Example 2 – Minimazation problem

In this example we have the fastest taxi company that has to assign each taxi to each passenger as fast as possible. The numbers in the matrix represent the time to reach the passenger.

We proceed as in the first example.

Iteration 1:

Now we have to assign the “0”s for every row respectively to the rule that we described earlier in example 1.

- In the first row we have one assignable “0” therefore we assign it and no other allocation in column 2 is possible.

- In the second row we have one assignable “0” therefore we assign it.

- In the third row we have several assignable “0”s. We leave it as it is for now and proceed.

- In the fourth and fifth row we have no assignable “0”s.

Now we proceed with the allocations of the “0”s for each column .

- In the first column we have one assignable “0” therefore we assign it. No other “0”s in row 3 are assignable anymore.

Now we are unable to proceed because all the “0”s either been assigned or crossed. The crosses indicate that they are not fit for assignments because assignments are already made.

We realize that we have 3 assignments for this 5x5 matrix. In the earlier example we were able to get 4 assignments for a 4x4 matrix. Now we have to follow another procedure to get the remaining 2 assignments (“0”).

Step 4 – Tick all unassigned rows.

Step 5 – If a row is ticked and has a “0”, then tick the corresponding column (if the column is not yet ticked).

Step 6 – If a column is ticked and has an assignment, then tick the corresponding row (if the row is not yet ticked).

Step 7 - Repeat step 5 and 6 till no more ticking is possible.

In this case there is no more ticking possible and we proceed with the next step.

Step 8 – Draw lines through unticked rows and ticked columns. The number of lines represents the maximum number of assignments possible.

Step 9 – Find out the smallest number which does not have any line passing through it. We call it Theta. Subtract theta from all the numbers that do not have any lines passing through them and add theta to all those numbers that have two lines passing through them. Keep the rest of them the same.

(With this step we create a new “0”)

With the new assignment matrix we start to assign the “0”s after the explained rules. Nevertheless we have 4 assignments against the required 5 for an optimal solution. Therefore we have to repeat step 4 – 9.

Iteration 2:

Step 4 – Tick all unassigned row.

Note: The indices of the ticks show you the order we added them.

Iteration 3:

Iteration 4:

After the fourth iteration we assign the “0”s again and now we have an optimal solution with 5 assignments.

The solution:

- Taxi1 => Passenger1 - duration 12

- Taxi2 => Passenger4 - duration 11

- Taxi3 => Passenger2 - duration 8

- Taxi4 => Passenger3 - duration 14

- Taxi5 => Passenger5 - duration 11

If we define the needed duration as costs, the minimal cost for this problem is 56.

Example 3 – Maximization problem

Furthermore the Hungarian algorithm can also be used for a maximization problem in which case we first have to transform the matrix. For example a company wants to assign different workers to different machines. Each worker is more or less efficient with each machine. The efficiency can be defined as profit. The higher the number, the higher the profit.

As you can see, the maximal profit of the matrix is 13. The simple twist that we do is rather than try to maximize the profit, we’re going to try to minimize the profit that you don’t get. If every value is taken away from 13, then we can minimize the amount of profit lost. We receive the following matrix:

From now on we proceed as usual with the steps to get to an optimal solution.

With the determined optimal solution we can compute the maximal profit:

- Worker1 => Machine2 - 9

- Worker2 => Machine4 - 11

- Worker3 => Machine3 - 13

- Worker4 => Machine1 - 7

Maximal profit is 40.

The optimal solution is found if there is one assigned “0” for each row and each column.

[1] Linear Assignment Problems and Extensions, Rainer E. Burkard, Eranda Cela

[2] Operations Research Skript TU Kaiserslautern, Prof. Dr. Oliver Wendt

[3] The Hungarian method for the assignment problem, H. W. Kuhn, Bryn Mawr College

Fundamental of Operations Research, Lec. 16, Prof. G. Srinivasan

Navigationsmenü

  • Quelltext anzeigen
  • Versionsgeschichte

Meine Werkzeuge

  • Gemeinschaftsportal
  • Operations Research
  • Studentenbeiträge zum Thema Operations Research
  • Wirtschaftsinformatik
  • Aktuelle Ereignisse
  • Letzte Änderungen
  • Zufällige Seite
  • Links auf diese Seite
  • Änderungen an verlinkten Seiten
  • Spezialseiten
  • Druckversion
  • Permanenter Link
  • Seiten­informationen

Powered by MediaWiki

  • Diese Seite wurde zuletzt am 1. Juli 2013 um 10:03 Uhr geändert.
  • Datenschutz
  • Über Operations-Research-Wiki

Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

Get full access to Quantitative Techniques: Theory and Problems and 60K+ other titles, with a free 10-day trial of O'Reilly.

There are also live events, courses curated by job role, and more.

HUNGARIAN METHOD

Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as Hungarian Method because of its special structure. If the time of completion or the costs corresponding to every assignment is written down in a matrix form, it is referred to as a Cost matrix. The Hungarian Method is based on the principle that if a constant is added to every element of a row and/or a column of cost matrix, the optimum solution of the resulting assignment problem is the same as the original problem and vice versa. The original cost matrix can be reduced to another cost matrix by adding constants to the elements of rows and columns where the total cost or the total completion time of an ...

Get Quantitative Techniques: Theory and Problems now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Don’t leave empty-handed

Get Mark Richards’s Software Architecture Patterns ebook to better understand how to design components—and how they should interact.

It’s yours, free.

Cover of Software Architecture Patterns

Check it out now on O’Reilly

Dive in for free with a 10-day trial of the O’Reilly learning platform—then explore all the other resources our members count on to build skills and solve problems every day.

write the steps of the hungarian method for assignment problems

Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

write the steps of the hungarian method for assignment problems

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

write the steps of the hungarian method for assignment problems

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

write the steps of the hungarian method for assignment problems

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

write the steps of the hungarian method for assignment problems

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

write the steps of the hungarian method for assignment problems

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

write the steps of the hungarian method for assignment problems

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

write the steps of the hungarian method for assignment problems

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

write the steps of the hungarian method for assignment problems

Column 3 contains no zero. Go to Step 2.

write the steps of the hungarian method for assignment problems

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

write the steps of the hungarian method for assignment problems

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

write the steps of the hungarian method for assignment problems

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

write the steps of the hungarian method for assignment problems

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

write the steps of the hungarian method for assignment problems

Step 3 (Assignment) :

write the steps of the hungarian method for assignment problems

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

write the steps of the hungarian method for assignment problems

The optimal assignment (minimum) cost = ₹ 35

Related Topics

Privacy Policy , Terms and Conditions , DMCA Policy and Compliant

Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.

IMAGES

  1. How to Solve an Assignment Problem Using the Hungarian Method

    write the steps of the hungarian method for assignment problems

  2. explain the steps in the hungarian method used for solving assignment

    write the steps of the hungarian method for assignment problems

  3. [#1]Assignment Problem[Easy Steps to solve

    write the steps of the hungarian method for assignment problems

  4. 12th Commerce

    write the steps of the hungarian method for assignment problems

  5. How to Solve Assignment Problem Hungarian Method- Simplest Way GATE Questions With Solutions

    write the steps of the hungarian method for assignment problems

  6. Assignment Problem (Part-2) Introduction to Hungarian Method to Solve

    write the steps of the hungarian method for assignment problems

VIDEO

  1. Assignment Problems-Hungarian Method Operation Research-04

  2. Assignment problem, simple probl

  3. ASSIGNMENT PROBLEM-TYPE II || HUNGARIAN METHOD || IN MALAYALAM

  4. Assignment Problem by Hungarian method, operations research

  5. | Fact Sheet

  6. Hungarian method in assignment problems minimization problems in HINDI

COMMENTS

  1. Hungarian Method

    Hungarian Method to Solve Assignment Problems The Hungarian method is a simple way to solve assignment problems. Let us first discuss the assignment problems before moving on to learning the Hungarian method. What is an Assignment Problem? A transportation problem is a type of assignment problem.

  2. Hungarian Algorithm for Assignment Problem

    cost matrix: 1500 4000 4500 2000 6000 3500 2000 4000 2500 Step 1: Subtract minimum of every row. 1500, 2000 and 2000 are subtracted from rows 1, 2 and 3 respectively. 0 2500 3000 0 4000 1500 0 2000 500 Step 2: Subtract minimum of every column. 0, 2000 and 500 are subtracted from columns 1, 2 and 3 respectively. 0 500 2500 0 2000 1000 0...

  3. Steps of the Hungarian Algorithm

    The Hungarian algorithm consists of four steps. The first two steps are executed once, while Steps 3 and 4 are repeated until an optimal assignment is found. The input of the algorithm is a square matrix with nonnegative elements.

  4. Hungarian Algorithm for Assignment Problem

    Step 1: Subtract minimum of every row. 2500, 3500 and 2000 are subtracted from rows 1, 2 and 3 respectively. 0 1500 1000 500 2500 0 0 2000 500 Step 2: Subtract minimum of every column. 0, 1500 and 0 are subtracted from columns 1, 2 and 3 respectively. 0 0 1000 500 1000 0

  5. PDF The Assignment Problem and the Hungarian Method

    The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to find an optimal assignment. Step 1. Subtract the smallest entry in each row from all the entries of its row. Step 2. Subtract the smallest entry in each column from all the entries of its column. Step 3.

  6. An Assignment Problem solved using the Hungarian Algorithm

    Step 1: Subtract row minima We start with subtracting the row minimum from each row. The smallest element in the first row is, for example, 69. Therefore, we substract 69 from each element in the first row. The resulting matrix is: Step 2: Subtract column minima

  7. PDF The Hungarian method for the assignment problem

    2. THE SIMPLE ASSIGNMENT PROBLEM The problem of Simple Assignment is illustrated by the following miniature example: Four individuals (denoted by i = 1, 2, 3, 4) are available for four - jobs (denoted by j = 1, 2, 3, 4). They qualify as follows:

  8. Hungarian algorithm

    Example In this simple example, there are three workers: Alice, Bob and Dora. One of them has to clean the bathroom, another sweep the floors and the third washes the windows, but they each demand different pay for the various tasks. The problem is to find the lowest-cost way to assign the jobs.

  9. Learn Hungarian Method

    Overview Test Series The Hungarian method, also known as the Kuhn-Munkres algorithm, is a computational technique used to solve the assignment problem in polynomial time. It's a precursor to many primal-dual methods used today. The method was named in honor of Hungarian mathematicians Dénes Kőnig and Jenő Egerváry by Harold Kuhn in 1955.

  10. The Hungarian Method for the Assignment Problem

    First Online: 01 January 2009 9310 Accesses 182 Citations 8 Altmetric Abstract This paper has always been one of my favorite "children," combining as it does elements of the duality of linear programming and combinatorial tools from graph theory. It may be of some interest to tell the story of its origin. Keywords Graph Theory

  11. Using the Hungarian Algorithm to Solve Assignment Problems

    Hungarian Algorithm Steps To use the Hungarian Algorithm, we first arrange the activities and people in a matrix with rows being people, columns being activity, and entries being the costs.

  12. Assignment Problem and Hungarian Algorithm

    The main idea of the method is the following: consider we've found the perfect matching using only edges of weight 0 (hereinafter called "0-weight edges"). Obviously, these edges will be the solution of the assignment problem.

  13. Hungarian Method Examples, Assignment Problem

    Example 1: Hungarian Method. The Funny Toys Company has four men available for work on four separate jobs. Only one man can work on any one job. The cost of assigning each man to each job is given in the following table. The objective is to assign men to jobs in such a way that the total cost of assignment is minimum. Job.

  14. How to Solve an Assignment Problem Using the Hungarian Method

    0:00 / 12:17 How to Solve an Assignment Problem Using the Hungarian Method Shokoufeh Mirzaei 17.1K subscribers Subscribe Subscribed 2.7K 237K views 6 years ago Linear Programming In this lesson...

  15. [PDF] The Hungarian method for the assignment problem

    A note on Hungarian method for solving assignment problem. Jayanta Dutta Subhas Chandra Pal. Computer Science, Mathematics. 2015. TLDR. Hungarian method is modified to find out the optimal solution of an assignment problem which reduces the computational cost of the method. Expand.

  16. Hungarian Method, Assignment Problem, Hungarian Algorithm

    The following steps summarize the approach: Steps in Hungarian Method 1. Identify the minimum element in each row and subtract it from every element of that row. 2. Identify the minimum element in each column and subtract it from every element of that column. 3.

  17. The Assignment Problem (Using Hungarian Algorithm)

    Step 6: Find the cost. Costs calculated using the previous table assignment. When we use the previous tables assignments of the jobs and cranes, we calculate the cost by adding the costs of the ...

  18. Assignment problem: Hungarian method 3

    The Hungarian method is a combinatorial optimization algorithm which was developed and published by Harold Kuhn in 1955. This method was originally invented for the best assignment of a set of persons to a set of jobs. It is a special case of the transportation problem. The algorithm finds an optimal assignment for a given "n x n" cost matrix.

  19. Hungarian Method to Assignment Problems

    This lecture explains the Hungarian method to find the optimal solution to the Assignment Problems. Other videos @DrHarishGarg Assignment Problem - Mathemati...

  20. Hungarian Method for Restriction on Assignment Problem-Example

    Solution. In the given problem there are 5 persons and 4 Jobs. The problem can be formulated as 5 × 4 5 × 4 assignment problem with cij c i j = time it to takes ith i t h person for jth j t h job. Let. xij = {1, 0, if ith Person is assigned to jth Job; otherwise. x i j = { 1, if i t h Person is assigned to j t h Job; 0, otherwise.

  21. Hungarian Method

    HUNGARIAN METHOD. Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as Hungarian Method because of its special structure. If the time of completion or the costs corresponding to every assignment is written down in a matrix form, it is referred to as a Cost matrix. The Hungarian Method is based on the principle that if a ...

  22. Hungarian Method for Unbalanced Assignment Problem-examples

    Step 1 In first row smallest is 32.9, second row is 33.1, third row is 28.5, fourth row is 26.4 and fifth row is 0. Subtract the minimum of each row of the above cost matrix, from all the elements of respective rows. The modified matrix is as follows: Assignment Problem Step 2 The minimum of each column of the modified matrix is 0.

  23. Solution of assignment problems (Hungarian Method)

    Step 1: Select a smallest element in each row and subtract this from all the elements in its row. Look for atleast one zero in each row and each column.Otherwise go to step 2. Step 2: Select the smallest element in each column and subtract this from all the elements in its column.