task assignment problem algorithm

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Job Assignment Problem using Branch And Bound

• N Queen Problem using Branch And Bound
• Traveling Salesman 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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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.

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 2023-01-02 UTC.

Multi-UAV Cooperative Task Assignment Algorithm Based on Heuristic Rules

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• First Online: 18 April 2024
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• Weiheng Liu 39 ,
• Sheng Cheng 39 ,
• Bo Jiang 39 &
• Ruyi Jiang 39  

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 1174))

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• International Conference on Autonomous Unmanned Systems

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With the increasingly complex battlefield environment and mission requirements, multi-UAV cooperative combat has become a hot research topic. The deceitful, sudden and hidden combat tasks in the complex battlefield environment put forward higher requirements for the autonomous decision-making ability and the application scope of the task assignment algorithm. For the multi-type and multi-constraint task assignment and reassignment problem, a solution idea is to integrate the knowledge and experience of human handling multi-UAV task assignment problem into the algorithm to improve the intelligence and feasibility of the solution algorithm. In this paper, based on the background of cooperative reconnaissance and attack integration of multiple UAVs in battlefield conditions and considering the temporal constraints of multiple tasks, a multi-type task assignment model is constructed for collaborative operations. On the basis of analyzing the advantages and disadvantages of existing solving algorithms, a heuristic rule-based multi-task assignment and reassignment method is proposed based on human knowledge and experience. Simulation results show that the proposed method is effective in solving multi-type, multi-demand and multi-constraint task assignment problems.

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Weiheng Liu, Sheng Cheng, Bo Jiang & Ruyi Jiang

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Liu, W., Cheng, S., Jiang, B., Jiang, R. (2024). Multi-UAV Cooperative Task Assignment Algorithm Based on Heuristic Rules. In: Qu, Y., Gu, M., Niu, Y., Fu, W. (eds) Proceedings of 3rd 2023 International Conference on Autonomous Unmanned Systems (3rd ICAUS 2023). ICAUS 2023. Lecture Notes in Electrical Engineering, vol 1174. Springer, Singapore. https://doi.org/10.1007/978-981-97-1091-1_40

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Title: multi-auv kinematic task assignment based on self-organizing map neural network and dubins path generator.

Abstract: To deal with the task assignment problem of multi-AUV systems under kinematic constraints, which means steering capability constraints for underactuated AUVs or other vehicles likely, an improved task assignment algorithm is proposed combining the Dubins Path algorithm with improved SOM neural network algorithm. At first, the aimed tasks are assigned to the AUVs by improved SOM neural network method based on workload balance and neighborhood function. When there exists kinematic constraints or obstacles which may cause failure of trajectory planning, task re-assignment will be implemented by change the weights of SOM neurals, until the AUVs can have paths to reach all the targets. Then, the Dubins paths are generated in several limited cases. AUV's yaw angle is limited, which result in new assignments to the targets. Computation flow is designed so that the algorithm in MATLAB and Python can realizes the path planning to multiple targets. Finally, simulation results prove that the proposed algorithm can effectively accomplish the task assignment task for multi-AUV system.

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The future of optimization: How 'Learn to Optimize' is reshaping algorithm design and configuration

O ptimization algorithms are pivotal in machine learning and artificial intelligence (AI) in general. For a long time, it has been widely believed that the design/configuration of optimization algorithms is a task that heavily relies on human intelligence and requires customized design for specific problems.

However, with the increasing demand for AI and the emergence of new and complex problems, the manual design paradigm is facing significant challenges. If machines can automatically or semi-automatically design optimization algorithms in some way, it will not only greatly alleviate these challenges but also substantially expand the horizons of AI.

In recent years, researchers have been exploring ways to automate the algorithm configuration and design process by learning from a set of training problem instances. These efforts, referred to as Learn to Optimize (L2O), utilize a large number of optimization problem instances as input and attempt to train optimization algorithms within a configuration space (or even code space) with generalization ability.

Results across fields such as SAT, machine learning, computer vision, and adversarial example generation have shown that the automatically/semi-automatically designed optimization algorithms can perform comparably to, or even outperform, manually designed ones. This suggests that the field of optimization algorithm design may have entered the dawn of "machine replacing human."

The article reviews three main approaches for L2O: training performance prediction models, training a single solver, and training a portfolio of solvers. It also discusses theoretical guarantees for the training process, successful application cases, and the generalization issues of L2O. Finally, this article points to promising future research directions.

The study is published in the journal National Science Review .

"L2O is expected to grow into a critical technology that relieves increasingly unaffordable human labor in AI." Tang says. However, he also points out that warranting reasonable generalization remains a challenge for L2O, especially when dealing with complex problem classes and solver classes.

"A second-stage fine-tuning might be necessary in many real-world scenarios," Tang suggests. "The learned solver(s) could be viewed as foundation models for further fine-tuning."

He believes that building a synergy between the training and fine-tuning of foundation models would be a critical direction for fully delivering the potential of L2O in future development.

More information: Ke Tang et al, Learn to optimize—a brief overview, National Science Review (2024). DOI: 10.1093/nsr/nwae132

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