Assignment Problem: Maximization
There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.
The Hungarian Method can also solve such assignment problems , as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss.
The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.
- Unbalanced Assignment Problem
- Multiple Optimal Solutions

Example: Maximization In An Assignment Problem
At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.
How should the counters be assigned to persons so as to maximize the profit ?
Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.
On small screens, scroll horizontally to view full calculation
Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.
Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.
Select the smallest element from all the uncovered elements, i.e., 4. Subtract this 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. Repeating step 3, we obtain a solution which is shown in the following table.
Final Table: Maximization Problem
Use Horizontal Scrollbar to View Full Table Calculation
The total cost of assignment = 1C + 2E + 3A + 4D + 5B
Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.
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Choose the correct alternative : To use the Hungarian method, a profit maximization assignment problem requires ______. - Mathematics and Statistics
Choose the correct alternative :
To use the Hungarian method, a profit maximization assignment problem requires ______.
Converting all profits to opportunity losses
A dummy person or job
Matrix expansion
Finding the maximum number of lines to cover all the zeros in the reduced matrix
Solution Show Solution
To use the Hungarian method, a profit maximization assignment problem requires converting all profits to opportunity losses .

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a. tan ( A − B ) = tan A − tan B \tan (A-B)=\tan A-\tan B tan ( A − B ) = tan A − tan B b. tan ( A − B ) = tan A − tan B 1 + tan A tan B \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} tan ( A − B ) = 1 + t a n A t a n B t a n A − t a n B
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Year, x Population, y 1950 151.33 1960 179.32 1970 203.30 1980 226.54 1990 248.72 2000 281.42 \begin{array}{|c|c|} \hline \text { Year, } \boldsymbol{x} & \text { Population, } \boldsymbol{y} \\ \hline 1950 & 151.33 \\ 1960 & 179.32 \\ 1970 & 203.30 \\ 1980 & 226.54 \\ 1990 & 248.72 \\ 2000 & 281.42 \\ \hline \end{array} Year, x 1950 1960 1970 1980 1990 2000 Population, y 151.33 179.32 203.30 226.54 248.72 281.42
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To use the hungarian method a profit-maximization assignment problem requires
To use the hungarian method a profit-maximization assignment problem requires.

The crosses indicate that they are not fit for assignments because assignments are already made. Step 2 — Subtract the column minimum from each column from the reduced matrix. In the optimal solution there should be only oneassignment in each row and columns of the given assignment table.
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. Should I just use “dummy” costs? The following example will explain it. Step 4 — Tick all unassigned row.
Worker 1 causes a cost of 6 for machine 1 and so on … To solve the problem we have to perform the following steps: We leave it as it is for now. Step 7 – Repeat step 5 and 6 till no more ticking is possible. Step 8 — Draw lines through unticked rows and ticked columns. Keep the rest of them the same. In this case there is no need to proceed any further steps. We leave it as it is for now and proceed. Step 4 An optimal assignment is found, if the number of assigned cells equals the number of rows and columns.
Step 9 — Find out the smallest number which does not have any line passing through it. The indices of the ticks show you the order we added them. The numbers in the matrix represent the time to reach the passenger.
It is a special case of the transportation problem. Each worker causes different costs for the machines. This method was originally invented for the best assignment of a set of persons to a set of jobs. The cost elements are givenand is a square matrix and requirement at each destination is one and availabilityat each origin is also one.
Therefore we have to repeat step 4 — 9. Step 4 — Tick all unassigned rows. 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. We realize that we have 3 assignments for this 5×5 matrix. Table Use Horizontal Scrollbar to View Full Table Calculation Job.

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Hungarian Method for Maximal Assignment Problem Examples
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Hungarian Method for Maximal Assignment Problem Example
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Profit per piece is Rs. 25. Find the maximum profit. Use the Hungarian method to determine the optimal assignments.
In the given problem there are 5 operators and 5 Lathe. The problem can be formulated as $5\times 5$ assignment problem with $c_{ij}$ = weekly output (in pieces) from $j^{th}$ Lathe by $i^{th}$ operator.
The profit per piece is Rs. 25. As the assignment problem is to maximize the profit, first we need to convert the assignment problem to minimization problem.
The minimum number of lines = 3, which is less than the order of assignment problem (i.e. 5). Hence the optimal assignment is not possible.
The smallest element in the matrix, not covered by the lines is 1. Subtract 1 from all the uncovered elements and add 1 at the intersection of horizontal and vertical lines. And obtain the second modified matrix.
Operator C $\to$ Lathe L3
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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:

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.
\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)
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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.
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- Discuss(20+)
- For each row of the matrix, find the smallest element and subtract it from every element in its row.
- Do the same (as step 1) for all columns.
- Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
- Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, 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.
Try it before moving to see the solution
Explanation for above simple example:
An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.
Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).
Space complexity : O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.
In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY This article is contributed by Yash Varyani . Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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To use the Hungarian method, a profit-maximization assignment problem requires a. converting all profits to opportunity losses.b. a dummy agent or task. c. matrix expansion.d. finding the maximum number of lines to cover all the zeros in the reduced matrix. ANS: A PTS: 1 TOP: Hungarian method a. converting all profits to opportunity losses .
The Hungarian Method can also solve such assignment problems, as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss. The conversion is accomplished by subtracting all the elements of the given matrix from the highest element.
Using Hungarian method the optimal assignment obtained for the following assignment problem to minimize the total cost is: 1 - D, 2 - A, 3 - B, 4 - C. VIEW SOLUTION Miscellaneous Exercise 7 | Q 1.09 | Page 127 Choose the correct alternative : The assignment problem is said to be unbalance if Number of rows is greater than number of columns
To use the Hungarian method, a profit maximization assignment problem requires converting all profits to opportunity losses. Concept: Hungarian Method of Solving Assignment Problem Is there an error in this question or solution? Chapter 7: Assignment Problem and Sequencing - Miscellaneous Exercise 7 [Page 126] Q 1.07 Q 1.06 Q 1.08 APPEARS IN
To use the Hungarian method, a profit-maximization assignment problem requires a. converting all profits to opportunity losses The transportation simplex method can be used to solve the assignment problem. T The transportation simplex method is limited to minimization problems F
Find Solution of Assignment problem using Hungarian method (MIN case) Solution: The number of rows = 5 and columns = 5 Here given problem is balanced. Step-1: Find out the each row minimum element and subtract it from that row Step-2: Find out the each column minimum element and subtract it from that column.
9. To use the Hungarian method, a profit-maximization assignment problem requires a. converting all profits to opportunity losses. b. a dummy agent or task.c. matrix expansion. d. finding the maximum number of lines to cover all the zeros in the reduced matrix. ANS: A PTS: 1 TOP: Hungarian method a. converting all profits to opportunity losses .
The Hungarian method for solving an assignment problem can also be used to solve; a. A transportation problem b. A travelling salesman problem c. Both (a) and (b) d. Only (b) An optimal solution of an assignment problem can be obtained only if; a. Each row and column has only one zero element b. Each row and column has at least one zero element c.
To use the hungarian method a profit-maximization assignment problem requires How To Solve An Assignment Problem. #1 This video is private The crosses indicate that they are not fit for assignments because assignments are already made. Step 2 — Subtract the column minimum from each column from the reduced matrix.
Here is the video about Maximization Assignment problem by using Hungarian method, in this video we have solve the problem by using simple step by step procedure which includes, Row...
Use the Hungarian method to determine the optimal assignments. Solution In the given problem there are 5 operators and 5 Lathe. The problem can be formulated as 5 × 5 assignment problem with cij = weekly output (in pieces) from jth Lathe by ith operator. Let xij = {1, if jth Lathe is assigned to ith Operator; 0, otherwise.
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.
The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal-dual methods.It was developed and published in 1955 by Harold Kuhn, who gave the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians: Dénes Kőnig and Jenő Egerváry.
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 …
Q2 List any three variations of assignment problem. Q3. To use the Hungarian method, a profit-maximization assignment problem requires I). Convening all profits to opportunity losses 2). A dummy agent or tack. 3). Matrix expansion 4). Finding the maximum number of lines to cover all the irons in the reduced metric Q4.
This is the video about Restricted assignment problemIn this we have solved restricted assignment problem with simple step by step procedure. Please try to w...
Q2 List any three variations of assignment problem. Q3. To use the Hungarian method, a profit-maximization assignment problem requires I). Convening all profits to opportunity losses 2). A dummy agent or tack. 3). Matrix expansion 4). Finding the maximum number of lines to cover all the irons in the reduced metric Q4.
Now use the Hungarian Method to solve the above problem. The maximum profit through this assignment is 214. Example 6. XYZ Ltd. employs 100 workers of which 5 are highly skilled workers that can be assigned to 5 technologically advanced machines. The profit generated by these highly skilled workers while working on different machines are as ...
Hungarian Algorithm Application. First, we want to turn our matrix into a square matrix by adding a dummy column with entries equal to 518 (the highest entry in the matrix). Now we have a 4 by 4 ...
In (1955), Kuhn developed the Hungarian method of the assignment problem, the reason for naming it with this name is because its basis liesby the effort of the Hungarian mathematician Egervàryin ...