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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.

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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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:

Distributed Algorithm Design for Constrained Multi-robot Task Assignment

The task assignment problem is one of the fundamental combinatorial optimization problems. It has been extensively studied in operation research, management science, computer science and robotics. Task assignment problems arise in various applications of multi-robot systems (MRS), such as environmental monitoring, disaster response, extraterrestrial exploration, sensing data collection and collaborative autonomous manufacturing. In these MRS applications, there are realistic constraints on robots and tasks that must be taken into account both from the modeling perspective and the algorithmic perspective. From the modeling aspect, such constraints include (a) Task group constraints: where tasks form disjoint groups and each robot can be assigned to at most one task in each group. One example of the group constraints comes from tightly-coupled tasks, where multiple micro tasks form one tightly-coupled macro task and need multiple robots to perform each simultaneously. (b) Task deadline constraints: where tasks must be assigned to meet their deadlines. (c) Dynamically-arising tasks: where tasks arrive dynamically and the payoffs of future tasks are unknown. Such tasks arise in scenarios like searchrescue, where new victims are found dynamically. (d) Robot budget constraints: where the number of tasks each robot can perform is bounded according to the resource it possesses (e.g., energy). From the solution aspect, there is often a need for decentralized solution that are implemented on individual robots, especially when no powerful centralized controller exists or when the system needs to avoid single-point failure or be adaptive to environmental changes. Most existing algorithms either do not consider the above constraints in problem modeling, are centralized or do not provide formal performance guarantees. In this thesis, I propose methods to address these issues for two classes of problems, namely, the constrained linear assignment problem and constrained generalized assignment problem. Constrained linear assignment problem belongs to P, while constrained generalized assignment problem is NP-hard. I develop decomposition-based distributed auction algorithms with performance guarantees for both problem classes. The multi-robot assignment problem is decomposed into an optimization problem for each robot and each robot iteratively solving its own optimization problem leads to a provably good solution to the overall problem. For constrained linear assignment problem, my approaches provides an almost optimal solution. For constrained generalized assignment problem, I present a distributed algorithm that provides a solution within a constant factor of the optimal solution. I also study the online version of the task allocation problem with task group constraints. For the online problem, I prove that a repeated greedy version of my algorithm gives solution with constant factor competitive ratio. I include simulation results to evaluate the average-case performance of the proposed algorithms. I also include results on multi-robot cooperative package transport to illustrate the approach.

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Mathematics > Optimization and Control

Title: learn to formulate: a surrogate model framework for generalized assignment problem with routing constraints.

Abstract: The generalized assignment problem with routing constraints, e.g. the vehicle routing problem, has essential practical relevance. This paper focuses on addressing the complexities of the problem by learning a surrogate model with reduced variables and reconstructed constraints. A surrogate model framework is presented with a class of surrogate models and a learning method to acquire parameters. The paper further provides theoretical results regarding the representational power and statistical properties to explore the effectiveness of this framework. Numerical experiments based on two practical problem classes demonstrate the accuracy and efficiency of the framework. The resulting surrogate models perform comparably to or surpass the state-of-the-art heuristics on average. Our findings provide empirical evidence for the effectiveness of utilizing size-reduced and reconstructed surrogate models in producing high-quality solutions.

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A Bi-Objective Constrained Robust Gate Assignment Problem: Formulation, Instances and Algorithm

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Joint decision-making for divisional seru scheduling and worker assignment considering process sequence constraints

• Original Research
• Published: 22 May 2024

Cite this article

• Lili Wang   ORCID: orcid.org/0000-0003-2308-7404 1 ,
• Min Li 1 , 2 ,
• Guanbin Kong 1 &
• Haiwen Xu   ORCID: orcid.org/0000-0002-0302-6439 2  

This paper concentrates on the divisional seru scheduling and worker assignment joint decision-making problem, and synthetically considers the difference in workers’ skill sets, the diversity of workers’ skill levels, the process sequence constraints, setup time, lot-splitting, etc., and then a nonlinear integer programming model is constructed to minimize the makespan . We show that it is necessary to consider the process sequence constraints, and the optimal makespan of the worker-operation allocation scheme without considering the process sequence constraints is much larger than that of considering the process sequence constraints. Moreover, as the number of workers increases, the advantage of considering sequence constraints becomes more obvious. Considering the multi-decision attributes and intractable computations of the studied problem, we turn it into bi-level programming. Then based on the combination of a hybrid genetic variable neighbourhood search algorithm (HGVNSA) and a greedy heuristic algorithm (GHA), a bi-level nested heuristic algorithm (HGVNSA-GHA) is designed. Finally, numerical experiment results show that the proposed algorithm can achieve better results and higher efficiency for the divisional seru scheduling and worker assignment model.

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Acknowledgements

This research was supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX22_0063), National Natural Science Foundation of China (12171236, 72394363), Natural Science Foundation of Jiangsu Province (BK20221435), and Sichuan Science and Technology Program (2023ZYD0010), CAAC Key Laboratory of Flight Techniques and Flight Safety (FZ2022ZZ05, FZ2022ZX60), Central University Basic Scientific Research Operation Cost Special Fund (J2021-057). We would like to give our great appreciation to all the reviewers and editors who contributed to this research.

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School of Management and Engineering, Nanjing University, Nanjing, 210093, China

Lili Wang, Min Li & Guanbin Kong

School of Science, Civil Aviation Flight University of China, Guanghan, 618307, China

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Wang, L., Li, M., Kong, G. et al. Joint decision-making for divisional seru scheduling and worker assignment considering process sequence constraints. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-05983-w

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Received : 12 June 2023

Accepted : 28 March 2024

Published : 22 May 2024

DOI : https://doi.org/10.1007/s10479-024-05983-w

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