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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

Step 3 (Assignment):

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

Column 3 contains no zero. Go to Step 2.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

Step 3 (Assignment) :

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

The optimal assignment (minimum) cost = ₹ 35

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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 )

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Assignment problem: Hungarian method 3

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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

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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 ...

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in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

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Artificial Intelligence and Evolutionary Algorithms in Engineering Systems pp 1–8 Cite as

Improvement in Hungarian Algorithm for Assignment Problem

  • Kartik Shah 5 ,
  • Praveenkumar Reddy 5 &
  • S. Vairamuthu 5  
  • Conference paper
  • First Online: 01 January 2014

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5 Citations

Part of the Advances in Intelligent Systems and Computing book series (AISC,volume 324)

Hungarian method for assignment problem is generally used in parallel environment for the assignment of job to a processor. If the number of processors and number of jobs are same, then we can assign each processor 1 job with less cost using Hungarian method. If the number of jobs is larger compared to number of processors, then this method does not work (another approach is using dummy processors, but it is not implementable). In this paper, we proposed an alternate approach same as Hungarian method for assignment of more jobs to lesser processors.

  • Assignment problem
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  • Improvement

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W. Kuhn, The hungarian method for assignment problem. Nav. Res. Logistics Q. 2 , 83–87 (1955)

Article   Google Scholar  

D.P. Bertsekas, Linear network optimization algorithms and codes (MIT Press, Cambridge, 1991)

MATH   Google Scholar  

G. Carpaneto, S. Martello, P. Toth, Algorithms and codes for assignment problem. Ann. Oper. Res. 13 , 193–223 (1988)

Article   MathSciNet   Google Scholar  

D.A. Castanon, B. Smith, A. Wilson, Performance of parallel assignment algorithms on different multiprocessor architectures . Alphatech report TP-1245, Burlington, MA, (1989)

Google Scholar  

U. Derigs, The shortest augmenting path method for solving assignment problem-motivation and computational experience. Ann. Oper. Res. 4 , 57–102 (1985)

M. Engqust, A successive shortest path algorithm for the assignment problem. INFRO. 20 , 370–384 (1982)

F. Glover, R. Glover, D. Klingman, Threshold assignment algorithm. Centre for Business Decision analysis report CBDA 107, Graduate School of Business, University of Taxes at Austin (1982)

J.R.M. Hall, An algorithm for distinct representatives. Am. Math. Monthly 51 , 716–717 (1956)

R. Jonkar, A. Volgenant, A shortest augmenting path algorithm for the dense and sparse linear assignment problems. Computing 3 , 92–106 (1987)

E. Lawler, Combinational Optimization: Networks and Matroids (Holt, Rinehart & Winston, New York, 1976), p. 206

L.F. Mcginnis, Implementation and testing of a primal dual algorithm for the assignment problem. Oper. Res. Int. J. 31 , 277–291 (1983)

MATH   MathSciNet   Google Scholar  

C.H. Papadimitriou, K. Steiglitz, Combinational Optimization: Algorithm and complexity (Prentice-Hall, Englewood Cliffs, 1982)

E. Balas, D. Miller, J. Pekny, P. Toth, A parallel shortest path algorithm for assignment problem. J. ACM 38 (4), 985–1004 (1991)

Article   MATH   MathSciNet   Google Scholar  

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Acknowledgments

The authors would like to thank the School of Computer Science and Engineering, VIT University, for giving them the opportunity to carry out this project and also for providing them with the requisite resources and infrastructure for carrying out the research.

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Bijaya Ketan Panigrahi

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Shah, K., Reddy, P., Vairamuthu, S. (2015). Improvement in Hungarian Algorithm for Assignment Problem. In: Suresh, L., Dash, S., Panigrahi, B. (eds) Artificial Intelligence and Evolutionary Algorithms in Engineering Systems. Advances in Intelligent Systems and Computing, vol 324. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2126-5_1

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Hungarian Method Examples

Now we will examine a few highly simplified illustrations of Hungarian Method for solving an assignment problem .

Later in the chapter, you will find more practical versions of assignment models like Crew assignment problem , Travelling salesman problem , etc.

Example-1, Example-2

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.

This is a minimization example of assignment problem . We will use the Hungarian Algorithm to solve this problem.

Identify the minimum element in each row and subtract it from every element of that row. The result is shown in the following table.

"A man has one hundred dollars and you leave him with two dollars, that's subtraction." -Mae West

On small screens, scroll horizontally to view full calculation

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

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, choose the cell arbitrarily for assignment.

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.

Use Horizontal Scrollbar to View Full Table Calculation

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 (ii) and (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.

Select the smallest element (i.e., 1) 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.

Now again make the assignments for the reduced matrix.

Final Table: Hungarian Method

Since the number of assignments is equal to the number of rows (& columns), this is the optimal solution.

The total cost of assignment = A1 + B4 + C2 + D3

Substituting values from original table: 20 + 17 + 17 + 24 = Rs. 78.

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in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

Hungarian Method to solve Assignment Problem

  • Jun 15, 2023

For obtaining an optimal assignment, Hungarian method involves following steps :

Subtract the minimum of each row of the cost matrix,from all the elements of respective rows.

Subtract the minimum of each column of the modified cost matrix, from all the elements of respective columns.

Then draw the minimum number of horizontal and vertical line to cover all the zeros in the modified cost matrix.

Let the number of lines be $N$.

  • If $N=n$, the number of rows (columns) of given cost matrix, then an optimal assignment can be made. Go to Step 6.
  • If $N<n$, then go to next step.

Determine the smallest element in the matrix, not covered by the $N$ lines. Subtract this smallest element from all the uncovered elements and add the same element at the intersection of horizontal and vertical lines. And obtain the second modified matrix.

Repeat Steps 3 and 4 until minimum number of lines become equal to the number of rows (columns) of the given matrix i.e. $N =n$.

Examine the row successively until a row-wise exactly single zero is found, mark this zero by $\square$ to make the assignment and mark cross $(\times)$ over all zeros in that column. Continue in this manner until all the rows have been examined. Repeat the same procedure for columns also.

Repeat the Step 6 successively until one of the following situations arise :

  • if no unmarked zero is left, the process ends; or
  • if there lie more than one of the unmarked zeros in any column or row, then mark $\square$ one of the unmarked zeros arbitrarily and cross over all zeros lying in that row or column. Repeat the process until no unmarked zero is left in the matrix.

Thus, in each row and in each column exactly one marked $\square$ zero is obtained. The assignment corresponding to these marked zeros will give an optimal assignment.

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

Assignment Problem: Meaning, Methods and Variations | Operations Research

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

IMAGES

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

    in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

    in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

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

    in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

  4. Hungarian Algorithm for Assignment Problem

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  5. JitenVaghela: Assignment Problem Solve by Hungarian Method with

    in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

  6. Assignment Problem (Part-3) Hungarian Method to solve Assignment

    in hungarian method of solving assignment problem the row opportunity cost matrix is obtained by

COMMENTS

  1. 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.

  2. PDF The Assignment Problem and the Hungarian Method

    cost matrix. 350 350 200 400 250 Let's look at one possible assignment. 250 400 200 400 600 400 350 350 250 The total cost of this assignment is $250 + $600 + $250 = $1100. Here's another possible assignment. 250

  3. Hungarian Method

    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.

  4. Hungarian Algorithm for Assignment Problem

    The Hungarian algorithm, aka Munkres assignment algorithm, utilizes the following theorem for polynomial runtime complexity ( worst case O (n3)) and guaranteed optimality: If a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an op...

  5. PDF A Critique of the Hungarian Method of Solving Assignment Problem to the

    1. Introduction The assignment problem arises because available resources (such as men, machines etc.) have varying degree of efficiency for performing different activity. ------------------------------------------------------------------------ * Corresponding author. Therefore, cost, profit or time of performing different activity is different.

  6. Hungarian Algorithm for Assignment Problem

    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.

  7. 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.

  8. Hungarian algorithm

    In the matrix formulation, we are given a nonnegative n × n matrix, where the element in the i -th row and j -th column represents the cost of assigning the j -th job to the i -th worker.

  9. (Pdf) Hungarian Algorithm to Solve the Assignment Problem With the Help

    HUNGARIAN ALGORITHM TO SOLVE THE ASSIGNMENT PROBLEM WITH THE HELP OF COMPUTIONAL TECHNIQUES ... Row opportunity cost matrix . ... New proposed method for solving Assignment problem And comparative ...

  10. Hungarian Method, Assignment Problem, Hungarian Algorithm

    Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way: For each row or column with a single zero value cell that has not be assigned or eliminated, box that zero value as an assigned cell. For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.

  11. 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 ...

  12. An Assignment Problem solved using the Hungarian Algorithm

    The total cost of this optimal assignment is to 69 + 37 + 11 + 23 = 140. Solve your own problem online. Four jobs need to be executed by four workers. A step by step explanation shows how the optimal assignment can be found using the Hungarian algorithm.

  13. 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. Once ...

  14. Improvement in Hungarian Algorithm for Assignment Problem

    Abstract. Hungarian method for assignment problem is generally used in parallel environment for the assignment of job to a processor. If the number of processors and number of jobs are same, then we can assign each processor 1 job with less cost using Hungarian method. If the number of jobs is larger compared to number of processors, then this ...

  15. Hungarian Method Examples, Assignment Problem

    Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way: For each row or column with a single zero value cell that has not be assigned or eliminated, box that zero value as an assigned cell. For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.

  16. PDF Assignment Problem Dr. Ramesh Kumar Chaturvedi

    The Hungarian method of solving the Assignment Problem Step 1: Determine the opportunity Cost table from Table 1. Subtract the lowest entry in each row with all the entries in that row. Subtract the lowest entry in a column in resulting table from a above with all entries in that column. The resultant Table is Opportunity cost table.

  17. (PDF) A Critique of the Hungarian Method of Solving Assignment Problem

    Assignment problems are concerned with finding a one-to-one distribution to achieve maximum profits and revenues or to reduce the cost or time to complete tasks, where the problem of assignment ...

  18. Hungarian method calculator

    Operation Research - Assignment problem calculator - Find solution of Assignment Problem Hungarian method, step-by-step online. ... Matrix & Vector: Numerical Methods: Statistical Methods: Operation Research: Word Problems: Calculus: Geometry: ... Hungarian method Type your data (either with heading or without heading), ...

  19. Modified Hungarian method for unbalanced assignment problem with

    This purpose can be served by assigning multiple jobs to a single machine. The present paper proposes a Modified Hungarian Method for solving unbalanced assignment problems which gives the optimal policy of assignment of jobs to machines. A stepwise algorithm of proposed method is presented and the developed algorithm is also coded in Java SE 11.

  20. Hungarian Method to solve Assignment Problem

    Hungarian Method to solve Assignment Problem. For obtaining an optimal assignment, Hungarian method involves following steps : Step 1. Subtract the minimum of each row of the cost matrix,from all the elements of respective rows. Step 2. Subtract the minimum of each column of the modified cost matrix, from all the elements of respective columns ...

  21. Learn Hungarian Method

    Steps of the Hungarian Method. The first step is to check if the problem is balanced i.e., the number of rows and columns are equal. If not, the problem needs to be balanced before applying the algorithm. Step 1: Subtract the smallest element in each row from all the elements of that row. Ensure that each row has at least one zero.

  22. PDF Using of Hungarian Method to Solve the Transportation Problems

    This type of problem can be solved by using the Hungarian Method and the algorithm of this method is based on finding the lowest cost in the case of minimization models and the greatest possible return in the case of maximization issues.

  23. Assignment Problem: Meaning, Methods and Variations

    Step 1: Develop the Cost Table from the given Problem: ADVERTISEMENTS: If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero. Step 2: Find the Opportunity Cost Table: