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.
\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)
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.
MATHS Related Links | |
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
Register with BYJU'S & Download Free PDFs
Register with byju's & watch live videos.
- Google OR-Tools
- Español – América Latina
- Português – Brasil
- Tiếng Việt
Solving an Assignment Problem
This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.
In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .
The costs of assigning workers to tasks are shown in the following table.
Worker | Task 0 | Task 1 | Task 2 | Task 3 |
---|---|---|---|---|
90 | 80 | 75 | 70 | |
35 | 85 | 55 | 65 | |
125 | 95 | 90 | 95 | |
45 | 110 | 95 | 115 | |
50 | 100 | 90 | 100 |
The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.
MIP solution
The following sections describe how to solve the problem using the MPSolver wrapper .
Import the libraries
The following code imports the required libraries.
Create the data
The following code creates the data for the problem.
The costs array corresponds to the table of costs for assigning workers to tasks, shown above.
Declare the MIP solver
The following code declares the MIP solver.
Create the variables
The following code creates binary integer variables for the problem.
Create the constraints
Create the objective function.
The following code creates the objective function for the problem.
The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.
Invoke the solver
The following code invokes the solver.
Print the solution
The following code prints the solution to the problem.
Here is the output of the program.
Complete programs
Here are the complete programs for the MIP solution.
CP SAT solution
The following sections describe how to solve the problem using the CP-SAT solver.
Declare the model
The following code declares the CP-SAT model.
The following code sets up the data for the problem.
The following code creates the constraints for the problem.
Here are the complete programs for the CP-SAT solution.
Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.
Last updated 2024-08-28 UTC.
- MapReduce Algorithm
- Linear Programming using Pyomo
- Networking and Professional Development for Machine Learning Careers in the USA
- Predicting Employee Churn in Python
- Airflow Operators
Solving Assignment Problem using Linear Programming in Python
Learn how to use Python PuLP to solve Assignment problems using Linear Programming.
In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.
If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.
The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.
In this tutorial, we are going to cover the following topics:
Assignment Problem
A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.
For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.
Problem Formulation
A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.
Initialize LP Model
In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.
Define Decision Variable
In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using LpVariable.dicts() class. LpVariable.dicts() used with Python’s list comprehension. LpVariable.dicts() will take the following four values:
- First, prefix name of what this variable represents.
- Second is the list of all the variables.
- Third is the lower bound on this variable.
- Fourth variable is the upper bound.
- Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are LpContinuous or LpInteger .
Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.
Define Objective Function
In this step, we will define the minimum objective function by adding it to the LpProblem object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.
Define the Constraints
Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem object.
Solve Model
In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.
From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.
In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in this article and learn about Linear Programming in this article .
- Solving Blending Problem in Python using Gurobi
- Transshipment Problem in Python Using PuLP
You May Also Like
Agglomerative Clustering
KNN Classification using Scikit-learn
Solving Multi-Period Production Scheduling Problem in Python using PuLP
Index Assignment problem Hungarian algorithm Solve online
Solve an assignment problem online
Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given.
Fill in the cost matrix ( random cost matrix ):
Don't show the steps of the Hungarian algorithm Maximize the total cost
HungarianAlgorithm.com © 2013-2024
The assignment problem revisited
- Original Paper
- Published: 16 August 2021
- Volume 16 , pages 1531–1548, ( 2022 )
Cite this article
- Carlos A. Alfaro ORCID: orcid.org/0000-0001-9783-8587 1 ,
- Sergio L. Perez 2 ,
- Carlos E. Valencia 3 &
- Marcos C. Vargas 1
1084 Accesses
4 Citations
4 Altmetric
Explore all metrics
First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the \(\epsilon \) - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.
This is a preview of subscription content, log in via an institution to check access.
Access this article
Subscribe and save.
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime
Price includes VAT (Russian Federation)
Instant access to the full article PDF.
Rent this article via DeepDyve
Institutional subscriptions
Similar content being viewed by others
Some results on an assignment problem variant
Integer Programming
A Full Description of Polytopes Related to the Index of the Lowest Nonzero Row of an Assignment Matrix
Bertsekas, D.P.: The auction algorithm: a distributed relaxation method for the assignment problem. Annal Op. Res. 14 , 105–123 (1988)
Article MathSciNet Google Scholar
Bertsekas, D.P., Castañon, D.A.: Parallel synchronous and asynchronous implementations of the auction algorithm. Parallel Comput. 17 , 707–732 (1991)
Article Google Scholar
Bertsekas, D.P.: Linear network optimization: algorithms and codes. MIT Press, Cambridge, MA (1991)
MATH Google Scholar
Bertsekas, D.P.: The auction algorithm for shortest paths. SIAM J. Optim. 1 , 425–477 (1991)
Bertsekas, D.P.: Auction algorithms for network flow problems: a tutorial introduction. Comput. Optim. Appl. 1 , 7–66 (1992)
Bertsekas, D.P., Castañon, D.A., Tsaknakis, H.: Reverse auction and the solution of inequality constrained assignment problems. SIAM J. Optim. 3 , 268–299 (1993)
Bertsekas, D.P., Eckstein, J.: Dual coordinate step methods for linear network flow problems. Math. Progr., Ser. B 42 , 203–243 (1988)
Bertsimas, D., Tsitsiklis, J.N.: Introduction to linear optimization. Athena Scientific, Belmont, MA (1997)
Google Scholar
Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. Revised reprint. SIAM, Philadelphia, PA (2011)
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18 (5), 1013–1036 (1989)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. Assoc. Comput. Mach. 35 , 921–940 (1988)
Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximation. Math. Op. Res. 15 , 430–466 (1990)
Goldberg, A.V., Kennedy, R.: An efficient cost scaling algorithm for the assignment problem. Math. Programm. 71 , 153–177 (1995)
MathSciNet MATH Google Scholar
Goldberg, A.V., Kennedy, R.: Global price updates help. SIAM J. Discr. Math. 10 (4), 551–572 (1997)
Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2 , 83–97 (1955)
Kuhn, H.W.: Variants of the Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2 , 253–258 (1956)
Lawler, E.L.: Combinatorial optimization: networks and matroids, Holt. Rinehart & Winston, New York (1976)
Orlin, J.B., Ahuja, R.K.: New scaling algorithms for the assignment ad minimum mean cycle problems. Math. Programm. 54 , 41–56 (1992)
Ramshaw, L., Tarjan, R.E., Weight-Scaling Algorithm, A., for Min-Cost Imperfect Matchings in Bipartite Graphs, : IEEE 53rd Annual Symposium on Foundations of Computer Science. New Brunswick, NJ 2012 , 581–590 (2012)
Zaki, H.: A comparison of two algorithms for the assignment problem. Comput. Optim. Appl. 4 , 23–45 (1995)
Download references
Acknowledgements
This research was partially supported by SNI and CONACyT.
Author information
Authors and affiliations.
Banco de México, Mexico City, Mexico
Carlos A. Alfaro & Marcos C. Vargas
Mountain View, CA, 94043, USA
Sergio L. Perez
Departamento de Matemáticas, CINVESTAV del IPN, Apartado postal 14-740, 07000, Mexico City, Mexico
Carlos E. Valencia
You can also search for this author in PubMed Google Scholar
Corresponding author
Correspondence to Carlos A. Alfaro .
Ethics declarations
Conflict of interest.
There is no conflict of interest.
Additional information
Publisher's note.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors were partially supported by SNI and CONACyT.
Rights and permissions
Reprints and permissions
About this article
Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4
Download citation
Received : 26 March 2020
Accepted : 03 August 2021
Published : 16 August 2021
Issue Date : June 2022
DOI : https://doi.org/10.1007/s11590-021-01791-4
Share this article
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
- Assignment problem
- Bertsekas auction algorithm
- Combinatorial optimization and matching
- Find a journal
- Publish with us
- Track your research
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.
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 30 | 37 | 40 | 28 | 40 |
2 | 40 | 24 | 27 | 21 | 36 |
3 | 40 | 32 | 33 | 30 | 35 |
4 | 25 | 38 | 40 | 36 | 36 |
5 | 29 | 62 | 41 | 34 | 39 |
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
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 32 | 25 | 22 | 34 | 22 |
2 | 22 | 38 | 35 | 41 | 26 |
3 | 22 | 30 | 29 | 32 | 27 |
4 | 37 | 24 | 22 | 26 | 26 |
5 | 33 | 0 | 21 | 28 | 23 |
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.
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 10 | 3 | 8 | ||
2 | 16 | 13 | 15 | 4 | |
3 | 8 | 7 | 6 | 5 | |
4 | 15 | 2 | 4 | ||
5 | 33 | 21 | 24 | 23 |
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
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 14 | 3 | 8 | ||
2 | 12 | 9 | 11 | ||
3 | 4 | 3 | 2 | 1 | |
4 | 19 | 2 | 4 | ||
5 | 37 | 21 | 24 | 23 |
The total cost of assignment = 1C + 2E + 3A + 4D + 5B
Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.
Share This Article
Operations Research Simplified Back Next
Goal programming Linear programming Simplex Method Transportation Problem
- Branch and Bound Tutorial
- Backtracking Vs Branch-N-Bound
- 0/1 Knapsack
- 8 Puzzle Problem
- Job Assignment Problem
- N-Queen Problem
- Travelling Salesman Problem
Job Assignment Problem using Branch And Bound
Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.
Let us explore all approaches for this problem.
Solution 1: Brute Force
We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).
Solution 2: Hungarian Algorithm
The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).
Solution 3: DFS/BFS on state space tree
A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.
Solution 4: Finding Optimal Solution using Branch and Bound
The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.
There are two approaches to calculate the cost function:
- For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
- For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).
In this article, the first approach is followed.
Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A.
Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red).
Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable.
Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14.
Below diagram shows complete search space diagram showing optimal solution path in green.
Complete Algorithm:
Below is the implementation of the above approach:
Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix. Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.
Similar Reads
- Branch and Bound
Please Login to comment...
- How to Watch NFL on NFL+ in 2024: A Complete Guide
- Best Smartwatches in 2024: Top Picks for Every Need
- Top Budgeting Apps in 2024
- 10 Best Parental Control App in 2024
- GeeksforGeeks Practice - Leading Online Coding Platform
Improve your Coding Skills with Practice
What kind of Experience do you want to share?
All Subjects
study guides for every class
That actually explain what's on your next test, assignment model, from class:, mathematical methods for optimization.
An assignment model is a type of optimization model used to determine the most efficient way to assign resources or tasks to agents in a way that minimizes costs or maximizes efficiency. This model typically involves finding the best one-to-one correspondence between two sets, such as jobs and workers, ensuring that each task is assigned to one agent and vice versa. Understanding the assignment model helps in solving various practical problems where resources must be allocated optimally.
congrats on reading the definition of assignment model . now let's actually learn it.
5 Must Know Facts For Your Next Test
- The assignment model is a special case of linear programming where the objective is to minimize costs associated with assigning tasks to agents.
- In a typical assignment problem, each agent can perform only one task, and each task must be assigned to one agent.
- The Hungarian algorithm is a widely used method for solving assignment problems efficiently, particularly when dealing with larger datasets.
- Assignment models can be applied in various fields such as workforce management, transportation logistics, and project scheduling.
- A feasible solution in an assignment model ensures that every task is completed while keeping the overall cost as low as possible.
Review Questions
- The assignment model optimizes resource allocation by ensuring that each task is assigned to a unique agent in a way that minimizes total costs or maximizes efficiency. Factors considered include the costs associated with each potential assignment, the availability of agents, and the specific requirements of each task. By analyzing these elements, the model finds the optimal pairing that achieves the best overall outcome for resource distribution.
- The Hungarian algorithm solves assignment problems by systematically reducing the cost matrix to find the minimum cost assignment while ensuring all agents are matched to tasks. Its advantages include efficiency in finding solutions quickly, even for larger problems, compared to brute force methods that can be computationally intensive. The algorithm also guarantees an optimal solution, making it a preferred choice for many practical applications.
- Using an assignment model in real-world scenarios can significantly improve efficiency and reduce costs in areas like workforce management and logistics. However, limitations include the assumption of linearity in relationships and potential constraints that may not be addressed by standard models. Additionally, factors like worker preferences, job satisfaction, or complex job requirements may complicate assignments beyond simple cost minimization. Thus, while effective, users must consider these nuances when applying the model in practice.
Related terms
Linear programming : A mathematical method used to determine the best outcome in a given mathematical model with constraints, often used in optimization problems including assignment models.
Hungarian algorithm : An efficient combinatorial optimization algorithm specifically designed to solve assignment problems by minimizing the total cost of assignments.
Bipartite graph : A type of graph where vertices can be divided into two distinct sets, and edges only connect vertices from different sets, often used in representing assignment problems.
" Assignment model " also found in:
© 2024 fiveable inc. all rights reserved., ap® and sat® are trademarks registered by the college board, which is not affiliated with, and does not endorse this website..
A discrete dwarf mongoose optimization algorithm to solve task assignment problems on smart farms
New citation alert added.
This alert has been successfully added and will be sent to:
You will be notified whenever a record that you have chosen has been cited.
To manage your alert preferences, click on the button below.
New Citation Alert!
Please log in to your account
Information & Contributors
Bibliometrics & citations, view options, index terms.
Computer systems organization
Architectures
Distributed architectures
Cloud computing
Computing methodologies
Distributed computing methodologies
Distributed algorithms
Power and energy
Energy distribution
Mathematics of computing
Discrete mathematics
Combinatorics
Combinatorial optimization
Mathematical analysis
Mathematical optimization
Discrete optimization
Network optimization
Network services
Theory of computation
Design and analysis of algorithms
Approximation algorithms analysis
Packing and covering problems
Recommendations
Approximation algorithm for the offloading problem in edge computing.
In the edge-cloud environment, offloading technique decides the task to be executed either at the cloud or at the edge. Offloading can improve the quality of service and the efficiency of the system. In most previous works on the offloading ...
Embedding Remaining Useful Life Predictions into a Modified Receding Horizon Task Assignment Algorithm to Solve Task Allocation Problems
The Task Allocation problem is one of the fundamental combinatorial optimization problems with applications on various domains. Solving a Task Allocation problem consists in, given a set of tasks to be performed and a set of resources, defining which ...
An Approximation Algorithm for Preemptive Scheduling on Parallel-Task Systems
This paper addresses the problem of preemptive scheduling on parallel-task systems. A parallel-task system consists of several independent tasks, each of which can be executed on one or more processors. Du and Leung introduced the problem of ...
Information
Published in.
Kluwer Academic Publishers
United States
Publication History
Author tags.
- Cloud-edge collaborative
- Unsplittable
- Approximation algorithm
- Dwarf mongoose algorithm
- Heavy tasks
- Research-article
Funding Sources
- This work was supported in part by the National Natural Science Foundation of China
Contributors
Other metrics, bibliometrics, article metrics.
- 0 Total Citations
- 0 Total Downloads
- Downloads (Last 12 months) 0
- Downloads (Last 6 weeks) 0
View options
Login options.
Check if you have access through your login credentials or your institution to get full access on this article.
Full Access
Share this publication link.
Copying failed.
Share on social media
Affiliations, export citations.
- Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download
- Download citation
- Copy citation
We are preparing your search results for download ...
We will inform you here when the file is ready.
Your file of search results citations is now ready.
Your search export query has expired. Please try again.
- Access through your organization
- Purchase PDF
Article preview
Introduction, section snippets, references (48).
Applied Soft Computing
An adaptive genetic algorithm with neighborhood search for integrated o2o takeaway order assignment and delivery optimization by e-bikes with varied compartments.
- • An integrated order assignment and delivery model with multi-type e-bikes is proposed.
- • An algorithm combining the genetic algorithm with neighborhood search is designed.
- • Numeric experiments using the instances simulating real-world features are performed.
- • Management insights for the takeaway delivery platform are provided.
O2O takeaway delivery problem
O2o takeaway order processing workflow, solution method, computational experiments, conclusions, credit authorship contribution statement, declaration of competing interest, acknowledgments, selling secondhand products through an online platform with blockchain, transp. res. e: logist. transp. rev., the role of numbers in the customer journey, the vehicle routing problem with backhauls, european j. oper. res., towards delivery-as-a-service: effective neighborhood search strategies for integrated delivery optimization of e-commerce and static o2o parcels, transp. res. b, emerging technology-based online scheduling for instant delivery in the o2o retail era, electron. commer. res. appl., investigate exact reliability under limited time and space of a multistate online food delivery network, expert syst. appl., crowdsourced on-demand food delivery: an order batching and assignment algorithm, transp. res. c, emerg. technol., optimisation of takeaway delivery routes considering the mutual satisfactions of merchants and customers, comput. ind. eng., evolutionary food quality and location strategies for restaurants in competitive online-to-offline food ordering and delivery markets: an agent-based approach, int. j. prod. econ., multi-compartment vehicle routing problems: state-of-the-art, modeling framework and future directions, on optimizing healthcare waste routing systems using waste separation policies: a case study, appl. soft. comput., a dynamic approach for the multi-compartment vehicle routing problem in waste management, renew. sust. energ. rev., solving the multi-compartment vehicle routing problem by an augmented lagrangian relaxation method, multi-compartment vehicle selection and delivery strategy with time windows under multi-objective optimization, alex. eng. j., a multi-compartment electric vehicle routing problem with time windows and temperature and humidity settings for perishable product delivery, a decision support system for consolidated distribution of a ceramic sanitary ware company, a multi-objective open vehicle routing problem with overbooking: exact and heuristic solution approaches for an employee transportation problem, cost-optimal deployment of autonomous mobile lockers co-operating with couriers for simultaneous pickup and delivery operations, transp. res. c, deploying autonomous mobile lockers in a two-echelon parcel operation, combining genetic local search into a multi-population imperialist competitive algorithm for the capacitated vehicle routing problem, a multi-depot vehicle routing problem with time windows, split pickup and split delivery for surplus food recovery and redistribution, evolutionary algorithm for vehicle routing for shared e-bicycle battery replacement and recycling, a novel two-phase approach for the bi-objective simultaneous delivery and pickup problem with fuzzy pickup demands, large neighbourhood search with adaptive guided ejection search for the pickup and delivery problem with time windows, eur. j. transp. logist., cited by (0).
IMAGES
VIDEO
COMMENTS
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 ...
The formal definition of the assignment problem (or linear assignment problem) is . Given two sets, A and T, together with a weight function C : A × T → R.Find a bijection f : A → T such that the cost function: (, ())is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as: , The problem is "linear" because the cost ...
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 it the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry.
Hungarian algorithm steps for minimization problem. Step 1: For each row, subtract the minimum number in that row from all numbers in that row. Step 2: For each column, subtract the minimum number in that column from all numbers in that column. Step 3: Draw the minimum number of lines to cover all zeroes.
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.
In this lesson we learn what is an assignment problem and how we can solve it using the Hungarian method.
The Hungarian method is a combinatorial optimization algorithm which solves the assignment problem in polynomial time . Later it was discovered that it was a primal-dual Simplex method.. It was developed and published by Harold Kuhn in 1955, who gave the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians: Denes Konig and Jeno ...
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.
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:
The following code creates the objective function for the problem. Python C++ Java C#. objective_terms = [] for i in range(num_workers): for j in range(num_tasks): objective_terms.append(costs[i][j] * x[i, j]) solver.Minimize(solver.Sum(objective_terms)) The value of the objective function is the total cost over all variables that are assigned ...
In this step, we will solve the LP problem by calling solve () method. We can print the final value by using the following for loop. From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.
General description of the algorithm. This problem is known as the assignment problem. The assignment problem is a special case of the transportation problem, which in turn is a special case of the min-cost flow problem, so it can be solved using algorithms that solve the more general cases. Also, our problem is a special case of binary integer ...
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.
Solve an assignment problem online. Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given. Fill in the cost matrix ():
THE HUNGARIAN METHOD FOR THE ASSIGNMENT. PROBLEM'. H. W. Kuhn. Bryn Y a w College. Assuming that numerical scores are available for the perform- ance of each of n persons on each of n jobs, the "assignment problem" is the quest for an assignment of persons to jobs so that the sum of the. n scores so obtained is as large as possible.
The first model for generating random bipartite graphs is the Erdös-Renyi model. Here, every possible edge of the bipartite graph has probability d to stay in the graph, ... New scaling algorithms for the assignment ad minimum mean cycle problems. Math. Programm. 54, 41-56 (1992) Article Google Scholar
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 ...
The Mathematical Model: Let ci,j be the cost of assigning the ith resource to the jth task. We define the cost matrix to be the n×n matrix C = c1,1 c1,2 ··· c1,n c2,1 c2,2 ··· c2,n..... cn,1 cn,2 ··· cn,n . An assignment is a set of n entry positions in the cost matrix, no two of which lie in the same row or column.
Solution 1: Brute Force. We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O (n!). Solution 2: Hungarian Algorithm. The optimal assignment can be found using the Hungarian algorithm.
The Hungarian algorithm is useful to identify minimum costs when people are assigned to specific activities based on cost. Practice using this algorithm in example equations of real-world scenarios.
An assignment model is a type of optimization model used to determine the most efficient way to assign resources or tasks to agents in a way that minimizes costs or maximizes efficiency. This model typically involves finding the best one-to-one correspondence between two sets, such as jobs and workers, ensuring that each task is assigned to one agent and vice versa.
Assignment Model: Hungarian Algorithm and its Applications. Assignment Problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum.
This paper presents a mathematical programming model and solution method for the path-constrained traffic assignment problem, in which route choices simultaneously follow the Wardropian equilibrium principle and yield the distance constraint imposed on the path.
In this study, we propose a novel cloud-edge collaborative task assignment model for smart farms that consists of a cloud server, m edge servers, and n sensors. The edge servers rely solely on solar-generated energy, which is limited, whereas the cloud server has access to a limitless amount of energy supplied by the smart grid.
An integrated order assignment and delivery model with multi-type e-bikes is proposed. An algorithm combining the genetic algorithm with neighborhood search is designed. Numeric experiments using the instances simulating real-world features are performed.