Assignment Problem: Meaning, Methods and Variations | Operations Research
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:
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.
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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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Assignment problem
The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :
maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $
$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$
(origins or supply),
$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$
(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.
If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).
In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.
The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.
In turn, the transportation problem is a special case of the network optimization problem.
A totally different assignment problem is the pole assignment problem in control theory.
[a1] | D.B. Yudin, E.G. Gol'shtein, "Linear programming" , Israel Program Sci. Transl. (1965) (In Russian) |
[a2] | R. Frisch, "La résolution des problèmes de programme linéaire par la méthode du potentiel logarithmique" , (1956) pp. 20–23 |
[a3] | K. Murtz, "Linear and combinatorial programming" , Wiley (1976) |
[a4] | M. Grötschel, L. Lovász, A. Schrijver, "Geometric algorithms and combinatorial optimization" , Springer (1987) |
[a5] | C.H. Papadimitriou, K. Steiglitz, "Combinatorial optimization" , Prentice-Hall (1982) |
- This page was last edited on 5 April 2020, at 18:48.
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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.
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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey
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WHAT IS ASSIGNMENT PROBLEM
Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.
The assignment problem in the general form can be stated as follows:
“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”
Several problems of management has a structure identical with the assignment problem.
Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...
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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.
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.
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.
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.
Since each row and column contains atleast one zero, assignments can be made.
Step 3 (Assignment):
Thus all the four assignments have been made. The optimal assignment schedule and total cost is
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.
∴ The given assignment problem is balanced.
Now let us find the solution.
The cost matrix of the given assignment problem is
Column 3 contains no zero. Go to Step 2.
Thus all the five assignments have been made. The Optimal assignment schedule and total cost is
The optimal assignment (minimum) cost = ` 9
Example 10.9
Solve the following assignment problem.
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
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.
Step 3 (Assignment) :
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
The optimal assignment (minimum) cost = ₹ 35
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Formation construction and reconfiguration control of UAV swarms: a perspective from distributed assignment and optimization
- Original Paper
- Published: 26 August 2024
Cite this article
- Jiangyuan Tian ORCID: orcid.org/0000-0002-4770-6524 1 ,
- Ruixuan Wei 2 &
- Longting Jiang 1
This article proposes a framework for the swarm of unmanned aerial vehicles (UAVs) to swiftly construct a formation. The framework also enables the swarm to reconfigure the formation effectively in a distributed way when the number of UAVs changes. Different from the classic formation methods, we endow each UAV with an objective function and an assignment vector. Then, a novel distributed optimization protocol is designed for the formation construction, which enables the swarm to construct the desired formation within a prescribed time. The formation needs to be reconfigured when some UAVs leave or join the swarm. A distributed assignment protocol is raised for the reconfiguration problem, which aims at reassigning the relative positions for the UAVs and reconfiguring the swarm formation. This article integrates these two protocols into one framework, where the transition between the formation construction and its reconfiguration is smooth. The distributed design of the framework improves the individual UAV’s decision-making ability. We further give the proofs of consensus and optimality of the two protocols. The simulation results confirm the effectiveness of the framework.
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Hayat, S., Yanmaz, E., Muzaffar, R.: Survey on unmanned aerial vehicle networks for civil applications: a communications viewpoint. IEEE Commun. Surv. Tutor. 18 (4), 2624–2661 (2016). https://doi.org/10.1109/COMST.2016.2560343
Article Google Scholar
Mozaffari, M., Saad, W., Bennis, M., Nam, Y.-H., Debbah, M.: A tutorial on UAVS for wireless networks: applications, challenges, and open problems. IEEE Commun. Surv. Tutor. 21 (3), 2334–2360 (2019). https://doi.org/10.1109/COMST.2019.2902862
Zhao, B., Xian, B., Zhang, Y., Zhang, X.: Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology. IEEE Trans. Ind. Electron. 62 (5), 2891–2902 (2015). https://doi.org/10.1109/TIE.2014.2364982
Wu, J., Luo, C., Luo, Y., Li, K.: Distributed UAV swarm formation and collision avoidance strategies over fixed and switching topologies. IEEE Trans. Cybern. 52 (10), 10969–10979 (2022). https://doi.org/10.1109/TCYB.2021.3132587
Shakhatreh, H., Sawalmeh, A.H., Al-Fuqaha, A., Dou, Z., Almaita, E., Khalil, I., Othman, N.S., Khreishah, A., Guizani, M.: Unmanned aerial vehicles (UAVS): a survey on civil applications and key research challenges. IEEE Access 7 , 48572–48634 (2019). https://doi.org/10.1109/ACCESS.2019.2909530
Iscold, P., Pereira, G.A.S., Torres, L.A.B.: Development of a hand-launched small UAV for ground reconnaissance. IEEE Trans. Aerosp. Electron. Syst. 46 (1), 335–348 (2010). https://doi.org/10.1109/TAES.2010.5417166
Wang, T., Li, M., Zhang, M.-Y.: Cooperative coverage reconnaissance of multi-UAV. In: 2020 IEEE 5th Information Technology and Mechatronics Engineering Conference (ITOEC), pp. 1647–1651. IEEE, Chongqing (2020). https://doi.org/10.1109/ITOEC49072.2020.9141873
Zhou, L., Leng, S., Liu, Q., Wang, Q.: Intelligent UAV swarm cooperation for multiple targets tracking. IEEE Internet Things J. 9 (1), 743–754 (2022). https://doi.org/10.1109/JIOT.2021.3085673
Wang, X., Zhou, W., Wang, X.: Research on multi-UAV cooperative tracking target based on PSO predictive control algorithm. IOP Conf. Ser. Mater. Sci. Eng. 646 (1), 012038 (2019). https://doi.org/10.1088/1757-899X/646/1/012038
Xu, Y., Ren, G., Chen, J., Zhang, X., Jia, L., Kong, L.: Interference-aware cooperative anti-jamming distributed channel selection in UAV communication networks. Appl. Sci. 8 (10), 1911 (2018). https://doi.org/10.3390/app8101911
Duan, H., Huo, M., Yang, Z., Shi, Y., Luo, Q.: Predator-prey pigeon-inspired optimization for UAV ALS longitudinal parameters tuning. IEEE Trans. Aerosp. Electron. Syst. 55 (5), 2347–2358 (2019). https://doi.org/10.1109/TAES.2018.2886612
Xing, D., Zhen, Z., Gong, H.: Offense-defense confrontation decision making for dynamic UAV swarm versus UAV swarm. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 233 (15), 5689–5702 (2019). https://doi.org/10.1177/0954410019853982
Liu, X., Ansari, N.: Resource allocation in UAV-assisted m2m communications for disaster rescue. IEEE Wirel. Commun. Lett. 8 (2), 580–583 (2019). https://doi.org/10.1109/LWC.2018.2880467
Ouyang, Q., Wu, Z., Cong, Y., Wang, Z.: Formation control of unmanned aerial vehicle swarms: a comprehensive review. Asian J. Control 25 (1), 570–593 (2023). https://doi.org/10.1002/asjc.2806
Kim, M.-H., Baik, H., Lee, S.: Response threshold model based UAV search planning and task allocation. J. Intell. Robot. Syst. 75 (3–4), 625–640 (2014). https://doi.org/10.1007/s10846-013-9887-6
Malvankar-Mehta, M.S., Mehta, S.S.: Optimal task allocation in multi-human multi-robot interaction. Optim. Lett. 9 (8), 1787–1803 (2015). https://doi.org/10.1007/s11590-015-0890-7
Article MathSciNet Google Scholar
Zhang, X., Duan, H.: An improved constrained differential evolution algorithm for unmanned aerial vehicle global route planning. Appl. Soft Comput. 26 , 270–284 (2015). https://doi.org/10.1016/j.asoc.2014.09.046
Liang, X., Meng, G., Luo, H., Chen, X.: Dynamic path planning based on improved boundary value problem for unmanned aerial vehicle. Clust. Comput. 19 (4), 2087–2096 (2016). https://doi.org/10.1007/s10586-016-0650-1
Lau, S.Y., Naeem, W.: Co-operative tensegrity-based formation control algorithm for a multi-aircraft system. In: 2015 American Control Conference (ACC), pp. 750–756. IEEE, Chicago (2015). https://doi.org/10.1109/ACC.2015.7170824
Liao, F., Teo, R., Wang, J.L., Dong, X., Lin, F., Peng, K.: Distributed formation and reconfiguration control of VTOL UAVs. IEEE Trans. Control Syst. Technol. 25 (1), 270–277 (2017). https://doi.org/10.1109/TCST.2016.2547952
Dong, X., Yu, B., Shi, Z., Zhong, Y.: Time-varying formation control for unmanned aerial vehicles: theories and applications. IEEE Trans. Control Syst. Technol. 23 (1), 340–348 (2015). https://doi.org/10.1109/TCST.2014.2314460
Li, N.H.M., Liu, H.H.T.: Formation UAV flight control using virtual structure and motion synchronization. In: 2008 American Control Conference, pp. 1782–1787. IEEE, Seattle (2008). https://doi.org/10.1109/ACC.2008.4586750
Xu, J., Guo, Q., Xiao, L., Chen, G., Guan, X.: An autonomous planning method for UAV based on behavior-conditional model. In: 2019 IEEE 7th International Conference on Computer Science and Network Technology (ICCSNT), pp. 255–261. IEEE, Dalian (2019). https://doi.org/10.1109/ICCSNT47585.2019.8962520
Ge, X., Han, Q.-L.: Distributed formation control of networked multi-agent systems using a dynamic event-triggered communication mechanism. IEEE Trans. Ind. Electron. 64 (10), 8118–8127 (2017). https://doi.org/10.1109/TIE.2017.2701778
You, K., Xie, L.: Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Trans. Autom. Control 56 (10), 2262–2275 (2011). https://doi.org/10.1109/TAC.2011.2164017
Yang, T., Yi, X., Wu, J., Yuan, Y., Wu, D., Meng, Z., Hong, Y., Wang, H., Lin, Z., Johansson, K.H.: A survey of distributed optimization. Annu. Rev. Control. 47 , 278–305 (2019). https://doi.org/10.1016/j.arcontrol.2019.05.006
Molzahn, D.K., Dorfler, F., Sandberg, H., Low, S.H., Chakrabarti, S., Baldick, R., Lavaei, J.: A survey of distributed optimization and control algorithms for electric power systems. IEEE Trans. Smart Grid 8 (6), 2941–2962 (2017). https://doi.org/10.1109/TSG.2017.2720471
Lin, Z., Liu, H.H.T.: Topology-based distributed optimization for multi-UAV cooperative wildfire monitoring: topology-based distributed optimization for multi-UAV cooperative wildfire monitoring. Opt. Control Appl. Methods 39 (4), 1530–1548 (2018). https://doi.org/10.1002/oca.2424
Rajasree, R., Jisha, V.R.: Optimal formation control of unmanned aerial vehicles with reconfiguration. In: 2015 International Conference on Control Communication & Computing India (ICCC), pp. 36–41. IEEE, Trivandrum (2015). https://doi.org/10.1109/ICCC.2015.7432866
Zhang, H., Zhao, G., Xu, G.: Time-optimal control for formation reconfiguration of multiple unmanned aerial vehicles. In: 2016 35th Chinese Control Conference (CCC), pp. 5630–5635. IEEE, Chengdu (2016). https://doi.org/10.1109/ChiCC.2016.7554234
Sperandio Giacomin, P.A., Hemerly, E.M.: Reconfiguration between longitudinal and circular formations for multi-UAV systems by using segments. J. Intell. Robot. Syst. 78 (2), 339–355 (2015). https://doi.org/10.1007/s10846-014-0063-4
Zhou, Y., Rao, B., Wang, W.: UAV swarm intelligence: recent advances and future trends. IEEE Access 8 , 183856–183878 (2020). https://doi.org/10.1109/ACCESS.2020.3028865
Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001). https://doi.org/10.1007/978-1-4613-0163-9
Book Google Scholar
Zhang, F.: Matrix Theory: Basic Results and Techniques. Springer, New York (2011). https://doi.org/10.1007/978-1-4614-1099-7
Li, T., Li, Y., Qian, Y.: Improved Hungarian algorithm for assignment problems of serial-parallel systems. J. Syst. Eng. Electron. 27 (4), 858–870 (2016). https://doi.org/10.21629/JSEE.2016.04.14
Ding, C., Shi, C., Chen, Y.: Nonsingular prescribed-time stabilization of a class of uncertain nonlinear systems: a novel coordinate mapping method. Int. J. Robust Nonlinear Control 30 (9), 3566–3581 (2020). https://doi.org/10.1002/rnc.4949
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This work was supported in part by Science and Technology Innovation 2030-Key Project of “New Generation Artificial Intelligence” under grant 2018AAA0102403. National Natural Science Foundation of China underGrant 61573373.
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Jiangyuan Tian & Longting Jiang
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Ruixuan Wei
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This appendix gives an extension of the simulation in the 3-dimensional scenario.
Although the result in the Fig. 16 illustrates that our method can make the UAVs track the mobile locating points and construct the formation within the prescribed time, the crossed trajectories in the 2-dimensional space still reflect that there is a latent risk of collisions between UAVs. In order to eliminate this risk, we extend the space to 3-dimension. The \(X-Y\) plane is for the formation mission and the introduction of the Z direction provides us with an additional dimension for collision avoidance between UAVs.
Trajectories of UAVs
Projection of UAVs
As is shown in the Fig. 17 , the locating points as well as the final formation are in the projection layer. The UAVs are initialized in different layers. Then the whole mission can be devided in 2 independent control problems: the control of the UAVs’ projections to construct the formation and the control of the Z coordinate to make the UAVs reach the projection layer. We have proposed our method for the first control problem in the previous article. Next we commence solving the second control problem.
Without causing any ambiguity, in this appendix we denote the Z coordinate of the i th UAV and the projection layer as \(z_i\) and \(z_d\) respectively. Then we propose a control protocol for the i th UAV to drive it to the projection layer:
Obviously, with the application of the protocol described in Eq. A1 , \(z_i\) will reach \(z_d\) as \(t\rightarrow \infty \) . Then we prove that the protocol described in Eq. A1 can also realize the collision avoidance between UAVs.
For \(\forall i,j\in \left\{ 1,2,\ldots ,n \right\} ,i\ne j\) , if \(z_i(0) > z_j(0)\) , then we have \(z_i(t) \ge z_j(t)\) for \(\forall t\ge 0\) and the equal sign is taken when \(t\rightarrow \infty \) .
Let \(F(t)=z_i(t)-z_j(t)\) . Take the derivative of F ( t ), we have:
Solve the differential equation A2 and take the initial condition into consideration, we have
It indicates that F ( t ) is a monotonic decreasing function and \(F(t) \ge 0\) for \(\forall t\ge 0\) . \(\square \)
The Theorem 3 implies that if the initial Z coordinate of the UAV i is greater than that of the UAV j , then during the whole formation process, the UAV i will be above the UAV j all the time until they both reach the projection layer in the end. We can realize the collision avoidance through guaranteeing that different UAVs will have different Z coordinates at the same time.
Trajectories of UAVs in 3-dimensional space
Then we give the simulation of the Sect. 4.3 in 3-dimensional space. All the simulation conditions, parameters and dynamic equations are the same as those in the Sect. 4.3 . The only difference is the Z coordinate needs to be added into the position information. The Z coordinate of the projection layer is set as 0. The initial Z coordinates of the 6 UAVs are set as 3, 2, \(-0.5\) , \(-1.5\) , \(-3\) , \(-3.5\) respectively. The simulation result is demonstrated in the Fig. 18 . From the result we can see that the trajectories of the UAVs don’t cross with each other. The Fig. 18 illustrates that with the addition of the Z direction and the application of the protocol described in Eq. A1 , the UAVs can construct the formation without the risk of collisions.
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Tian, J., Wei, R. & Jiang, L. Formation construction and reconfiguration control of UAV swarms: a perspective from distributed assignment and optimization. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-10024-z
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Accepted : 10 July 2024
Published : 26 August 2024
DOI : https://doi.org/10.1007/s11071-024-10024-z
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IMAGES
COMMENTS
Worked example of assigning tasks to an unequal number of workers using the Hungarian method. The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent ...
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 ...
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 ...
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.
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform as many tasks as possible by assigning at most one ...
If it is perfect, then the problem is solved. Otherwise find the minimum vertex cover V (for the subgraph with 0-weight edges only), the best way to do this is to use Köning's graph theorem. Step 2) Let and adjust the weights using the following rule: Step 3) Repeat Step 1 until solved.
Exercise 17 shows that the number of iterations is O(n2). To compare the Hungarian method to the exhaustive search method mentioned above, suppose that each iteration can be performed in one second. Then an assignment prob-lem with n = 30 can be solved in at most 302 = 900 seconds, or 15 minutes of computer time.
Assignment problems deal with the question of how to assign other items (machines, tasks). There are different ways in mathematics to describe an assignment: we can view an assignment as a bijective mapping φ between two finite sets . We can write a permutation φ as 12…nφ (1)φ (2)…φ (n), which means that 1 is mapped to φ (1), 2 is ...
The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $. If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).
The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the ...
The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to find an optimal assignment. Step 1. Subtract the smallest entry in each row from all the entries of its row. Step 2. Subtract the smallest entry in each column from all the entries of its column. Step 3.
Problem 4. Job shop needs to assign 4 jobs to 4 workers. The cost of performing a job is a function of the skills of the workers. Table summarizes the cost of the assignments. Worker1 cannot do job3, and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method. Job.
1.. IntroductionAlthough the name "assignment problem" seems to have first appeared in a 1952 paper by Votaw and Orden [69], what is generally recognized to be the beginning of the development of practical solution methods for and variations on the classic assignment problem (hereafter referred to as the AP) was the publication in 1955 of Kuhn's article on the Hungarian method for its ...
Connection Between Transportation and Assignment Problem An assignment problem is a special case of transportation problem in which m = n, all a i and b j are unity and each is limited to either 0 or 1. Hungarian Method for Solving an Assignment Problem 1. Prepare a square n n matrix. If not, make it square by adding suitable number of dummy ...
Definition of Assignment Problem. The statement of the assignment problem is as follows: There are n men and n jobs, with a cost c, for assigning man i to job j. It is required to assign all men to jobs such that one and only one man is assigned to each job and the total cost of the assignments is minimal.
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.
Assignment Problems 7 Hungarian Method of Solving an Assignment Problem The steps for obtaining an optimal solution of an assignment problem are as follows: 1. Check whether the given matrix is square. If not, make it square by adding a suitable number of dummy rows (or columns) with 0 cost/time elements. 2.
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):
An assignment problem is a particular case of transportation problem. The objective is to assign a number of resources to an equal number ... PROBLEM 1: Solve the following assignment problem shown in Table using Hungarian method. The matrix entries are processing time of each man in hours. 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I II III ...
UNIT -2. r: IIASSIGNMENT PROBLEMIntroduction:Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a. number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of ...
The Hungarian assignment method problem, developed by mathematician D. Konig, is a faster and more efficient approach to solving assignment problems. It involves determining the cost of making all possible assignments using a matrix. Each problem has a row representing the objects to be assigned and columns representing assigned tasks.
Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to ...
The revised assignment problem is. 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. Step 2 : Step 3 (Assignment) : 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.
This article proposes a distributed assignment and optimization method for the formation construction and reconfiguration control of the UAV swarm. Based on the classical Hungarian algorithm for the assignment problem, we remodel the problem in a distributed way and design an distributed assignment protocol described in Eq. 12. By virtue of the ...