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:
for i = 1, 2, ..., n and j = 1, 2, ..., n
In the above table, c ij is the cost of performing jth job by ith worker.
Generalized Form of an Assignment Problem
The optimization model is
Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn
subject to x i1 + x i2 +..........+ x in = 1 i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1 j = 1, 2,......., n
x ij = 0 or 1
In Σ Sigma notation
x ij = 0 or 1 for all i and j
An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.
Assumptions in Assignment Problem
- Number of jobs is equal to the number of machines or persons.
- Each man or machine is assigned only one job.
- Each man or machine is independently capable of handling any job to be done.
- Assigning criteria is clearly specified (minimizing cost or maximizing profit).
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Min max solution to the random assignment problem
Consider the standard assignment problem : $n$ people are assigned to $n$ jobs (one person to one job) so to minimize the sum of costs. When the costs are generated randomly (using the exponential (1) distribution), it is known that the expected sum of costs is $\pi^2/6$ .
An alternative solution to the assignment problem is to choose an allocation that instead minimizes the largest costs incurred by any individual, in the spirit of the John Rawls fairness criteria (see here ). Is it known any algorithm that finds such maximin allocation, and is it known what is the random sum of costs in the random version of the problem?
Many thanks,
- assignment-problem
Yes, there is an efficient algorithm to find an assignment that minimizes the largest cost. Suppose we want to check whether there is an assignment with largest cost $t$ . To do that, delete all edges with cost larger than $t$ , ignore the costs, and check whether the graph has a bipartite matching using the Hopcroft-Karp algorithm; if it does, there is a valid assignment. Now use binary search over $t$ until you find the minimum $t$ such that a valid assignment exists. Thus, the problem can be solved $O(|E| \sqrt{|V|} \lg |E|)$ time.
- $\begingroup$ Thank you. I have asked the second question here cs.stackexchange.com/questions/140571/… , I would appreciate any comments on it $\endgroup$ – fox May 21, 2021 at 9:55
- $\begingroup$ Are you familiar with any paper that implements such an algorithm or studies this problem? $\endgroup$ – fox May 21, 2021 at 9:56
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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
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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.
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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
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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
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Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods
Table of Contents
Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.
What is the Unbalanced Assignment Problem?
The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.
Formulation of the Unbalanced Assignment Problem
To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.
Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.
Solutions for the Unbalanced Assignment Problem
Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.
In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.
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Operations Research
1 Operations Research-An Overview
- History of O.R.
- Approach, Techniques and Tools
- Phases and Processes of O.R. Study
- Typical Applications of O.R
- Limitations of Operations Research
- Models in Operations Research
- O.R. in real world
2 Linear Programming: Formulation and Graphical Method
- General formulation of Linear Programming Problem
- Optimisation Models
- Basics of Graphic Method
- Important steps to draw graph
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- Solving Linear Programming Graphically Using Computer
- Application of Linear Programming in Business and Industry
3 Linear Programming-Simplex Method
- Principle of Simplex Method
- Computational aspect of Simplex Method
- Simplex Method with several Decision Variables
- Two Phase and M-method
- Multiple Solution, Unbounded Solution and Infeasible Problem
- Sensitivity Analysis
- Dual Linear Programming Problem
4 Transportation Problem
- Basic Feasible Solution of a Transportation Problem
- Modified Distribution Method
- Stepping Stone Method
- Unbalanced Transportation Problem
- Degenerate Transportation Problem
- Transhipment Problem
- Maximisation in a Transportation Problem
5 Assignment Problem
- Solution of the Assignment Problem
- Unbalanced Assignment Problem
- Problem with some Infeasible Assignments
- Maximisation in an Assignment Problem
- Crew Assignment Problem
6 Application of Excel Solver to Solve LPP
- Building Excel model for solving LP: An Illustrative Example
7 Goal Programming
- Concepts of goal programming
- Goal programming model formulation
- Graphical method of goal programming
- The simplex method of goal programming
- Using Excel Solver to Solve Goal Programming Models
- Application areas of goal programming
8 Integer Programming
- Some Integer Programming Formulation Techniques
- Binary Representation of General Integer Variables
- Unimodularity
- Cutting Plane Method
- Branch and Bound Method
- Solver Solution
9 Dynamic Programming
- Dynamic Programming Methodology: An Example
- Definitions and Notations
- Dynamic Programming Applications
10 Non-Linear Programming
- Solution of a Non-linear Programming Problem
- Convex and Concave Functions
- Kuhn-Tucker Conditions for Constrained Optimisation
- Quadratic Programming
- Separable Programming
- NLP Models with Solver
11 Introduction to game theory and its Applications
- Important terms in Game Theory
- Saddle points
- Mixed strategies: Games without saddle points
- 2 x n games
- Exploiting an opponent’s mistakes
12 Monte Carlo Simulation
- Reasons for using simulation
- Monte Carlo simulation
- Limitations of simulation
- Steps in the simulation process
- Some practical applications of simulation
- Two typical examples of hand-computed simulation
- Computer simulation
13 Queueing Models
- Characteristics of a queueing model
- Notations and Symbols
- Statistical methods in queueing
- The M/M/I System
- The M/M/C System
- The M/Ek/I System
- Decision problems in queueing
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Minimizing maximum cost in an one-to-one assignment problem
I want to compute a one-to-one assignment, from A to B where A and B have n elements. Each element of A will be assigned to one element in B. There is an associated cost between each element of A and B. The goal is to minimize the maximum cost of the assignment.
How to compute such an assignment using the cost matrix? I need to select an element from each row/column such that minimax strategy is achieved.
I was able to achieve this by listing all possible assignments and calculating their maximum costs and then selecting the one with the least maximum cost, but I wonder if there's a better approach.
- optimization
- game-theory
- You're looking for a "minimum cost matching in a complete balanced bipartite graph" -- searching using these terms should give you good results. bipartite means you have 2 sets of vertices with edges between them. Balanced means these sets are the same size. Complete means that all edges between the two sets are present. A matching is a set of edges s.t. each vertex is on at most one edge. – Dave Dec 15, 2022 at 16:44
- homes.di.unimi.it/righini/Didattica/OttimizzazioneCombinatoria/… – Dave Dec 15, 2022 at 20:35
2 Answers 2
The easiest is probably as a MIP (Mixed-Integer Programming) model:
This can be solved with any MIP solver.
A brute force polynomial cost algorithm based on repeated application of max matching in bipartite graph:
Order the pairs by assignment cost. Take n cheapest pairs. This determines a bipartite graph with A and B as vertices (pairs are edges). Try to find a matching in this graph. If found, it determines the optimal one-to-one assignment.
If not found, add the next pair. Repeat until a matching is found.
I suspect this can be optimized by not running the matching algorithm from the start every time you add an edge but rather keeping the maximum matching and checking if it can be augmented with the newly added edge.
- If you squint, this is sort of what the Hungarian algorithm is doing, but the extra complexity in manipulating the dual solution is what makes it correct. – David Eisenstat Dec 16, 2022 at 14:44
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One of the most well-known combinatorial optimization problems is the assignment problem . Here's an example: suppose a group of workers needs to perform a set of tasks, and for each worker and task, there is a cost for assigning the worker to the task. 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.
You can visualize this problem by the graph below, in which there are four workers and four tasks. The edges represent all possible ways to assign workers to tasks. The labels on the edges are the costs of assigning workers to tasks.
An assignment corresponds to a subset of the edges, in which each worker has at most one edge leading out, and no two workers have edges leading to the same task. One possible assignment is shown below.
The total cost of the assignment is 70 + 55 + 95 + 45 = 265 .
The next section shows how solve an assignment problem, using both the MIP solver and the CP-SAT solver.
Other tools for solving assignment problems
OR-Tools also provides a couple of other tools for solving assignment problems, which can be faster than the MIP or CP solvers:
- Linear sum assignment solver
- Minimum cost flow solver
However, these tools can only solve simple types of assignment problems. So for general solvers that can handle a wide variety of problems (and are fast enough for most applications), we recommend the MIP and CP-SAT solvers.
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.
IMAGES
VIDEO
COMMENTS
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.
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 relative penalties associated with assigning resources to an activity as opposed to making the best or least cost assignment. If we can reduce the cost matrix to the extent of having at least one zero in each row and ...
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.
Cost = 95. Worker 4 assigned to task 5. Cost = 45. Time = 0.000245 seconds. The result is the same as that for the linear assignment solver (except for the different numbering of workers and costs). The linear assignment solver is slightly faster than min cost flow — 0.000147 seconds versus 0.000458 seconds.
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.
A step-by-step demonstration on how to calculate the minimum cost (cost minimization)of an assignment in an Assignment Problem in Operations Research or Mana...
assignment problem occurs frequently in practice and is a basic problem in network flow theory since it can be reduced to a number of other problems, including the shortest path, weighted matching, transportation, and minimal cost flow [4]. ... QAP aims to reduce total assignment costs. Adding resource placement costs and
The mathematical model for the classic assignment problem may be given as: Minimize ∑ i = 1 n ∑ j = 1 n c ij x ij Subject to: ∑ i = 1 n x ij = 1 j = 1, …, n, ∑ j = 1 n x ij = 1 i = 1, …, n, x ij = 0 or 1, where x ij = 1 if agent i is assigned to task j, 0 if not, and c ij = the cost of assigning agent i to task j. The first set of ...
Section 7.5: The Assignment Problem In this section, we investigate the assignment problem- That is, given n jobs and n people, assign every job to a unique person. Typically, there are either costs or time involved, and we would want to make the assignments in such a way as to minimize this quantity.
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.
In an assignment problem, the goal is to assign n tasks to n agents in such a way that the overall cost or benefit is minimized or maximized. The maximization problem arises when the objective is to maximize the overall benefit rather than minimize the cost. Understanding Maximisation in an Assignment Problem. The maximization problem can be ...
The Balanced Assignment Problem (BAP) is a variant of the classic AP where instead of minimizing the total cost, we minimize the max-min distance which is the difference between the maximum assignment cost and the minimum one in the assignment solution. In [ 2 ], the authors proposed an efficient threshold-based algorithm to solve the BAP in ...
The Generalized Assignment Problem 1.1 Introduction The generalized assignment problem (GAP) asks to assign n clients to m servers in such a way that the assignment cost is minimized, provided that all clients are assigned to a server and that thecapacityof each server is not exceeded. A mathematical model can be obtainedby defining:1
The assignment problem is a special type of transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.. In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.
When the costs are generated randomly (using the exponential (1) distribution), it is known that the expected sum of costs is π2/6 π 2 / 6 . An alternative solution to the assignment problem is to choose an allocation that instead minimizes the largest costs incurred by any individual, in the spirit of the John Rawls fairness criteria (see ...
In the \(\epsilon \)-scaling auction algorithm every object v has a price p(v).This set of prices can be seen as a price function over V defined by \(p:V\rightarrow {\mathbb {R}}\).For every edge \(uv\in E\) we define the reduced cost \(w'(uv)\) of job v for the person u as \(w(uv) - p(v)\).. The objective in the auction algorithm is to find prices p(v) for the jobs and a perfect matching M ...
The Assignment Problem: An Example. The Assignment Problem: An Example. A company has 4 machines available for assignment to 4 tasks. Any machine can be assigned to any task, and each task requires processing by one machine. The time required to set up each machine for the processing of each task is given in the table below.
The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has ...
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. In an ... The Simplex Algorithm is a general purpose algorithm to find the maximum/minimum of a linear cost function with linear constraints.
1. A brute force polynomial cost algorithm based on repeated application of max matching in bipartite graph: Order the pairs by assignment cost. Take n cheapest pairs. This determines a bipartite graph with A and B as vertices (pairs are edges). Try to find a matching in this graph. If found, it determines the optimal one-to-one assignment.
We first introduce the given information for the optimal assignment problem. (1) The CFG G is shown in Fig. 1(b). (2) As for the plants, the production cost and delay are given in Fig. 2. (3) As for the storage cost, for each unit at each plant, the cost is $10. Based on this, the cost of initial storage units for each product is $ 12.
The total cost of the assignment is 70 + 55 + 95 + 45 = 265. The next section shows how solve an assignment problem, using both the MIP solver and the CP-SAT solver. Other tools for solving assignment problems. OR-Tools also provides a couple of other tools for solving assignment problems, which can be faster than the MIP or CP solvers:
Solving The Assignment Problems Directly Without Any Iterations Mai Mismar Al-Quds Open University ... Suppose that an assignment problem has n machines and n jobs so as to minimize the total cost or time in such a way that each machine can be assigned to one and only one job [1],[3] and [6]. The cost matrix C ij is given as: 11 12 1