## Cost matrix: How to construct and use it in assignment problems

1. introduction to assignment problems, 2. understanding the cost matrix, 3. constructing the cost matrix, 4. interpreting the cost matrix, 5. solving assignment problems using the cost matrix, 6. optimizing the cost matrix, 7. real-life applications of assignment problems, 8. limitations and challenges of cost matrix, 9. conclusion and further resources.

The process of assigning tasks or duties to a group of individuals or machines can be a daunting and time-consuming task. This is especially true when there are multiple tasks to be completed with different requirements and varying levels of expertise required. However, there is a solution to this problem: the assignment problem. The assignment problem is a fundamental problem in optimization theory and operations research that deals with the optimization of the allocation of resources to different tasks. In this section, we will introduce the assignment problem, discuss its importance and applications, and provide an overview of the different methods used to solve it.

1. Definition of Assignment Problem: The assignment problem is a combinatorial optimization problem that involves finding the best matching between two sets of items. In the context of resource allocation, it involves assigning a set of tasks to a group of individuals or machines with the objective of minimizing the total cost or maximizing the total profit.

2. Applications of Assignment Problem: The assignment problem has a wide range of applications in different fields such as transportation, logistics, scheduling, and economics. For example, in transportation, the problem can be used to assign delivery routes to trucks with the objective of minimizing the total distance traveled or the total time taken.

3. cost matrix : The cost matrix is a key component in solving the assignment problem. It is a matrix that represents the cost or profit of assigning each task to each individual or machine. The rows of the matrix represent the tasks, while the columns represent the individuals or machines. The values in the matrix represent the cost or profit of assigning each task to each individual or machine.

4. Methods to Solve Assignment Problems: There are several methods to solve the assignment problem, including the Hungarian algorithm, the shortest path algorithm, and the auction algorithm. Each method has its own advantages and disadvantages, and the choice of method depends on the specific problem and the available resources.

The assignment problem is a fundamental problem in optimization theory and operations research that deals with the optimization of resource allocation. The cost matrix is a key component in solving the problem, and there are several methods available to solve it. Understanding the assignment problem and its applications is essential for any organization that needs to allocate resources efficiently .

Introduction to Assignment Problems - Cost matrix: How to construct and use it in assignment problems

When it comes to solving assignment problems , cost matrix plays a crucial role in finding the optimal solution. The cost matrix is a table that contains the cost of assigning a particular task to a particular agent. Understanding the cost matrix is essential to construct and use it effectively in assignment problems. From a mathematical perspective, the cost matrix represents the objective function, which is the function that needs to be optimized. From a practical perspective, the cost matrix represents the real-world constraints and limitations.

Here are some in-depth insights on understanding the cost matrix:

1. The cost matrix can be asymmetric, which means the cost of assigning a task to an agent may not be the same as the cost of assigning the same task to another agent. For example, let's say we have three tasks and three agents. The cost of assigning task 1 to agent 1 may be different from the cost of assigning task 1 to agent 2. This asymmetry can occur due to several factors, such as skill level, availability, and distance.

2. The cost matrix can have infinite values, which means some tasks cannot be assigned to certain agents. For example, let's say we have four tasks and two agents. Task 1 and task 2 require a particular skill that agent 1 does not possess. Therefore, the cost of assigning task 1 or task 2 to agent 1 is infinite. This infinite value indicates that assigning these tasks to agent 1 is not feasible.

3. The cost matrix can include negative values, which means some tasks are profitable to assign. For example, let's say we have two tasks and two agents. Assigning task 1 to agent 1 results in a profit of $10, whereas assigning task 2 to agent 2 results in a profit of $5. In this case, the cost matrix includes negative values, which means we can maximize the profit by assigning tasks to the appropriate agents.

Understanding the cost matrix is crucial to construct and use it effectively in assignment problems. By considering the insights mentioned above, we can create an accurate cost matrix that reflects the real-world constraints and limitations.

Understanding the Cost Matrix - Cost matrix: How to construct and use it in assignment problems

When it comes to solving assignment problems, constructing the cost matrix is an essential step. The cost matrix represents the costs of assigning a certain worker to a specific task. The matrix can be constructed in different ways, depending on the type of the assignment problem. There are two main types of assignment problems: the balanced assignment problem, where the number of workers is equal to the number of tasks, and the unbalanced assignment problem, where the number of workers is not equal to the number of tasks. Each type of assignment problem requires a different approach in constructing the cost matrix.

Here are some insights about constructing the cost matrix:

1. For the balanced assignment problem, the cost matrix is a square matrix, where the number of rows and columns are equal to the number of workers or tasks. The cost of assigning a worker to a task is placed in the corresponding cell of the matrix. For example, consider a scenario where there are three workers and three tasks. The cost matrix would look like this:

| | Task 1 | Task 2 | Task 3 |

|Worker 1| 2 | 5 | 1 |

|Worker 2| 3 | 2 | 6 |

|Worker 3| 4 | 1 | 7 |

2. For the unbalanced assignment problem, the cost matrix is a rectangular matrix, where the number of rows and columns are not equal. In this case, a dummy row or column is added to make the matrix square. The cost of assigning a worker to a task is placed in the corresponding cell of the matrix, and the cost of assigning a worker to the dummy task or a task to the dummy worker is set to infinity. For example, consider a scenario where there are three workers and two tasks. The cost matrix would look like this:

| | Task 1 | Task 2 | Dummy Task |

|Worker 1| 2 | 5 | |

|Worker 2| 3 | 2 | |

|Worker 3| 4 | 1 | |

3. The cost matrix can also be constructed using distance measures, such as Euclidean distance or Manhattan distance. In this case, the cost represents the distance between the worker and the task. For example, consider a scenario where the workers and tasks are located in a two-dimensional space. The cost matrix can be constructed using the Euclidean distance between the worker and the task. The cost of assigning a worker to a task is then placed in the corresponding cell of the matrix.

Constructing the cost matrix is a crucial step in solving assignment problems. It determines the cost of assigning a worker to a task and provides the necessary information for finding the optimal solution. Different types of assignment problems require different approaches in constructing the cost matrix. By following the insights provided above, one can construct the cost matrix effectively and efficiently.

Constructing the Cost Matrix - Cost matrix: How to construct and use it in assignment problems

The cost matrix is an essential component in assignment problems. It is a square matrix that contains the costs of assigning each element in the row to each element in the column. The cost matrix can be interpreted in different ways, depending on the problem at hand. In this section, we will discuss how to interpret the cost matrix in assignment problems.

1. Interpretation as a distance matrix

One way to interpret the cost matrix is as a distance matrix. This is particularly useful when the assignment problem involves finding the shortest distance between two points. In this case, the cost matrix would contain the distances between each point. For example, consider a delivery company that needs to find the shortest route between several locations. The cost matrix would contain the distances between each location, and the assignment problem would involve finding the shortest route that visits each location once.

2. Interpretation as a similarity matrix

Another way to interpret the cost matrix is as a similarity matrix. This is useful when the assignment problem involves finding the best match between two sets of objects. In this case, the cost matrix would contain the similarities between each object. For example, consider a dating app that needs to match users based on their interests. The cost matrix would contain the similarities between each user's interests, and the assignment problem would involve finding the best match for each user.

3. Interpretation as a cost matrix

The most common interpretation of the cost matrix is as a cost matrix . This is useful when the assignment problem involves minimizing the total cost of assigning elements from one set to another. In this case, the cost matrix would contain the costs of assigning each element in the row to each element in the column. For example, consider a company that needs to assign employees to projects. The cost matrix would contain the costs of assigning each employee to each project, and the assignment problem would involve minimizing the total cost of assigning all employees to a project.

The cost matrix can be interpreted in different ways depending on the problem at hand. It can be interpreted as a distance matrix, a similarity matrix, or a cost matrix. Understanding the different interpretations of the cost matrix is crucial in constructing and solving assignment problems.

Interpreting the Cost Matrix - Cost matrix: How to construct and use it in assignment problems

When solving an assignment problem, the cost matrix plays a crucial role. It helps to identify the optimal assignment, where the sum of the costs is minimized. The cost matrix is constructed using the costs associated with each task and each agent. Assigning a task to an agent results in a cost that is entered into the matrix. When the matrix is complete, optimization algorithms can be used to identify the optimal assignment. Here are some key points to keep in mind when solving assignment problems using the cost matrix:

1. The cost matrix should be constructed with care. It is important to ensure that each cost is accurately reflected in the matrix. One way to do this is to double-check the costs before entering them into the matrix. For example, if the cost of assigning task A to agent X is $5, make sure that this cost is correct before entering it into the matrix.

2. The optimal assignment can be identified using optimization algorithms. The Hungarian algorithm and the auction algorithm are two common algorithms used to find the optimal assignment. These algorithms take the cost matrix as input and output the optimal assignment.

3. The optimal assignment may not always be unique. In some cases, there may be multiple assignments that result in the same minimum cost. When this happens, any of the optimal assignments can be chosen.

4. The cost matrix can be used to solve a variety of assignment problems. For example, it can be used to assign tasks to employees, to assign vehicles to routes, and to assign machines to production tasks.

5. The cost matrix can be modified to reflect changing costs. If the cost of assigning a task to an agent changes, the cost matrix can be updated accordingly. This allows for the optimal assignment to be recalculated based on the new costs.

The cost matrix is an essential tool for solving assignment problems. It allows for the identification of the optimal assignment, which results in minimized costs. By constructing the matrix carefully and using optimization algorithms, the optimal assignment can be found quickly and accurately.

Solving Assignment Problems using the Cost Matrix - Cost matrix: How to construct and use it in assignment problems

optimizing the cost matrix is an essential step in constructing and using it in assignment problems. This process aims to minimize the overall cost of the assignment, making it an efficient and cost-effective solution. The optimization process can be approached from different points of view, such as minimizing the total cost, maximizing the overall efficiency, or achieving a balance between cost and efficiency. In this section, we will discuss some of the methods used to optimize the cost matrix and their advantages.

1. Row and column reduction: This method involves subtracting the minimum value of each row and column from the cost matrix to reduce the overall cost. The advantage of this method is that it is simple and easy to implement. For example, suppose we have a cost matrix of size n x n. To reduce the cost, we need to find the minimum value of each row and subtract it from all the elements in that row. Similarly, we need to find the minimum value of each column and subtract it from all the elements in that column.

2. Hungarian algorithm: This algorithm is an efficient method for solving assignment problems and optimizing the cost matrix. It involves finding the minimum number of lines required to cover all the zeros in the reduced matrix. The advantage of this method is that it guarantees an optimal solution and is suitable for large-sized matrices. For example, suppose we have a cost matrix of size n x n. To use the Hungarian algorithm, we need to reduce the matrix using the row and column reduction method and then find the minimum number of lines required to cover all the zeros in the reduced matrix.

3. Branch and bound method: This method involves dividing the problem into sub-problems and solving them recursively. The advantage of this method is that it can handle complex problems and find the optimal solution. For example, suppose we have a cost matrix of size n x n. To use the branch and bound method, we need to divide the problem into sub-problems and solve them recursively until we find the optimal solution.

Optimizing the cost matrix is an important step in constructing and using it in assignment problems. The methods discussed above are some of the most commonly used methods to optimize the cost matrix and find the optimal solution. Depending on the problem's complexity and size, one method may be more suitable than the others.

Optimizing the Cost Matrix - Cost matrix: How to construct and use it in assignment problems

When it comes to real-life applications , the assignment problem is frequently utilized in various fields. One of the most common areas is in logistics, where the problem is used to optimize transportation of goods. The assignment problem can also be applied in sports, where it can help coaches determine which team members should play in a particular game. Additionally, the assignment problem can be used in the field of computer science , where it is used to solve problems related to resource allocation. Here are some more specific examples:

1. In the healthcare industry, the assignment problem can be used to match patients with the most appropriate healthcare providers . This can help ensure that patients receive the care they need from the most qualified professionals.

2. In the field of economics, the assignment problem can be used to match workers with jobs. This can help to minimize unemployment and ensure that the right people are in the right jobs.

3. In the field of agriculture, the assignment problem can be used to determine which crops should be grown on which fields. By optimizing crop selection, farmers can improve yields and maximize profits.

Overall, the assignment problem has a wide range of applications and can be used to solve a variety of different problems. Whether you are working in logistics, healthcare, economics, or agriculture, the assignment problem can help you optimize your operations and achieve better results.

Real life Applications of Assignment Problems - Cost matrix: How to construct and use it in assignment problems

The cost matrix is an essential component of assignment problems. It provides a visual representation of the costs associated with assigning tasks to individuals or machines. However, like any other tool, the cost matrix has its limitations and challenges that one should be aware of when using it. These limitations and challenges can affect the accuracy and efficiency of the results obtained from the cost matrix. In this section, we will discuss some of the limitations and challenges of the cost matrix.

1. Difficulty in determining costs: One of the primary challenges of the cost matrix is determining the costs associated with assigning tasks. Depending on the nature of the task, the costs can be challenging to estimate. For example, if the task involves a machine, the cost could be the energy consumed, maintenance required, or depreciation of the machine. On the other hand, if the task involves an individual, the cost could be the hourly wage, training costs, or travel expenses. Determining these costs can be time-consuming, and if not done accurately, it can lead to inaccurate results.

2. Limited flexibility: The cost matrix assumes that the cost of each task is fixed and does not change over time. However, in real-world scenarios , the cost of tasks can vary depending on several factors, such as demand, availability, and competition. Therefore, the cost matrix may not be suitable for situations where the cost of tasks is highly variable.

3. Inability to account for qualitative factors: The cost matrix only considers quantitative factors, such as the cost of tasks and resources. It does not account for qualitative factors such as the skills and experience of individuals or the quality of the equipment used. Therefore, the cost matrix may not be suitable for situations where the quality of work is critical.

4. Size limitations: The size of the cost matrix can also be a limitation. As the number of tasks and resources increases, the size of the cost matrix can become too large to handle efficiently. This can lead to longer computation times, increased memory requirements, and reduced accuracy.

5. Limited use cases: The cost matrix is most suitable for assignment problems where the number of tasks and resources is relatively small. In situations where the number of tasks and resources is significant, alternative optimization techniques such as linear programming may be more appropriate.

While the cost matrix is a valuable tool for solving assignment problems, it is essential to be aware of its limitations and challenges. By understanding these limitations and challenges, we can make better decisions about when and how to use the cost matrix.

Limitations and Challenges of Cost Matrix - Cost matrix: How to construct and use it in assignment problems

Constructing and using cost matrices is an important part of solving assignment problems. From a mathematical perspective, cost matrices provide a clear representation of the costs associated with each assignment. From a practical standpoint, cost matrices allow businesses and organizations to optimize their resources and assign tasks in the most efficient way possible.

If you're interested in learning more about cost matrices and their applications, there are a number of resources available. Here are a few worth checking out:

1. Linear programming textbooks - Cost matrices are an important part of linear programming, and many textbooks on the subject cover them in detail. Some popular options include "Linear Programming: Foundations and Extensions" by Robert Vanderbei and "Introduction to Linear Optimization" by Dimitris Bertsimas and John N. Tsitsiklis.

2. Online tutorials - If you prefer to learn through online resources, there are a number of tutorials available that cover cost matrices. For example, the website MathIsFun.com has a clear and concise tutorial on the subject.

3. Problem sets - The best way to learn how to use cost matrices is to practice with them. Look for problem sets online or create your own using real-world scenarios. For example, imagine you run a catering business and need to assign tasks to your employees for a large event. Use a cost matrix to determine the most efficient way to assign tasks based on each employee's skills and availability.

By familiarizing yourself with cost matrices and their applications, you'll be better equipped to solve assignment problems and optimize resources in a variety of settings.

Conclusion and Further Resources - Cost matrix: How to construct and use it in assignment problems

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## 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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## 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
- Multiple, Unbounded Solution and Infeasible Problems
- 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

## 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:

Use matchpairs to match the points at t = 0 with the points at t = 1 . To do this, first calculate a cost matrix where C(i,j) is the Euclidean distance from point i to point j .

Next, use matchpairs to match the rows and columns in the cost matrix. Specify the cost of unassignment as 1. With such a low cost of unassignment relative to the entries in the cost matrix, it is likely matchpairs will leave some points unmatched.

The values M(:,2) correspond to the original points ( x 0 , y 0 ) , while the values M(:,1) correspond to the moved points ( x 1 , y 1 ) .

Plot the matched pairs of points. The points that moved farther than 2*costUnmatched away from the original point remain unmatched.

## Input Arguments

Cost — cost matrix matrix.

Cost matrix. Each entry Cost(i,j) specifies the cost of assigning row i to column j .

Data Types: single | double

## costUnmatched — Cost of not matching scalar

Cost of not matching, specified as a scalar. matchpairs compares the value of 2*costUnmatched to the entries in Cost to determine whether it is more beneficial for a row or column to remain unmatched. Use this parameter to make matches more or less likely in the algorithm. For more information, see linear assignment problem .

Example: M = matchpairs(C,10) specifies a cost of 10 for not matching a row or column of C .

## goal — Optimization goal 'min' (default) | 'max'

Optimization goal, specified as either 'min' or 'max' . The optimization goal specifies whether the total cost should be minimized or maximized.

Example: M = matchpairs(Cost,costUnmatched,'max') specifies that the rows and columns of Cost should be matched together to maximize the total cost.

## Output Arguments

M — matches matrix.

Matches, returned as a matrix. M is a p -by- 2 matrix, where M(i,1) and M(i,2) are the row and column indices of a matched pair in the cost matrix. The rows of M are sorted with the second column in ascending order.

Each row and column can be matched a single time only, so each M(i,1) value and each M(i,2) value is unique.

M contains p matches, and p is less than or equal to the maximum number of matches min(size(Cost)) .

The cost of the matches in M is sum([Cost(M(1,1),M(1,2)), Cost(M(2,1),M(2,2)), ..., Cost(M(p,1),M(p,2))]) .

## uR — Unassigned rows column vector

Unassigned rows, returned as a column vector of indices. The entries in uR indicate which rows in Cost are unassigned. Each entry in uR and uC contributes to the total cost of the solution according to costUnassigned .

## uC — Unassigned columns column vector

Unassigned columns, returned as a column vector of indices. The entries in uC indicate which columns in Cost are unassigned. Each entry in uR and uC contributes to the total cost of the solution according to costUnassigned .

## Linear Assignment Problem

The linear assignment problem is a way of assigning rows to columns such that each row is assigned to a column and the total cost of the assignments is minimized (or maximized). The cost of assigning each row to each column is captured in a cost matrix . The entry Cost(i,j) is the cost of assigning row i to column j .

The cost of unassignment assigns a cost to any row or column that is not matched. This practice allows for minimum-cost solutions that do not assign all rows or columns. If a row and column are not matched, this increases the total cost by 2*costUnmatched .

The total cost of a solution M is the sum of the cost of all matched pairs added to the cost of all unmatched pairs:

T C = ∑ i = 1 p Cost ( M ( i , 1 ) , M ( i , 2 ) ) + costUnmatched ⋅ ( m + n − 2 p )

In code the total cost is

Cost is an m -by- n matrix.

M is a p -by- 2 matrix, where M(i,1) and M(i,2) are the row and column of a matched pair.

(m+n-2*p) is the total number of unmatched rows and columns.

[1] Duff, I.S. and J. Koster. "On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix." SIAM J. Matrix Anal. & Appl. 22(4), 2001. pp 973–996.

## Extended Capabilities

C/c++ code generation generate c and c++ code using matlab® coder™..

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.

## Version History

Introduced in R2019a

equilibrate | sprank | dmperm

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## Assignment Problem

- Feb 20, 2024

The assignment problem is a special case of transportation problem.

Suppose there are $n$ jobs to be performed and $n$ persons are available for these jobs. Assume that each person can do each job at a time, though with varying degree of efficiency.

Let $c_{ij}$ be the cost if the $i^{th}$ person is assigned the $j^{th}$ job. Then the problem is to find an assignment so that the total cost of performing all jobs is minimum. (i.e., which job should be assigned to which person with minimum cost). Such problem is called an Assignment Problem (AP).

The tabular form of the assignment problem is as follows

The above table is called the $n\times n$ cost-matrix, where $c_{ij}$ are real numbers.

Thus the objective in the Assignment Problem is to assign a number of jobs to the equal number of persons at a minimum cost or maximum profit. e.g. assigning men to offices, classes to rooms, drivers to trucks, problems to research teams etc.

## Mathematical Formulation of AP

Mathematically, the AP can be stated as $$ \begin{equation}\label{eq2.4} \min z = \sum_{i=1}^n\sum_{j=1}^n x_{ij} c_{ij}. \end{equation} $$ subject to $$ \begin{equation}\label{eq2.5} \sum_{j=1}^n x_{ij} =1,; \text{ for } i=1,2,\ldots, n \end{equation} $$

$$ \begin{equation}\label{eq2.6} \sum_{i=1}^n x_{ij} =1,; \text{ for } j=1,2,\ldots,n \end{equation} $$ where $$ \begin{equation*} x_{ij}=\left{ \begin{array}{ll} 1, & \hbox{if $i^{th}$ person is assigned $j^{th}$ job;} \ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$ Constraint \eqref{eq2.5} indicate that one job is done by $i^{th}$ person $i=1,2,\ldots, n$ and constraint \eqref{eq2.6} indicate that one person should be assigned $j^{th}$ job $j=1,2,\ldots, n$.

It may be observed from the above formulation that AP is a special type of Linear programming problem.

## Unbalanced Assignment Problem

If the cost matrix of an assignment problem is not a square matrix, the assignment problem is called an Unbalanced Assignment Problem . In such a case, add dummy row or dummy column with zero cost in the cost matrix so as to form a square matrix. Then apply the usual assignment method to find the optimal solution.

## Maximal Assignment Problem

Sometimes, the assignment problem deals with the maximization of an objective function instead of minimization.. For example, it may be required to assign persons to jobs in such a way that the expected profit is maximum. In such a case, first convert the problem of maximization to minimization and apply the usual procedure of assignment.

The assignment problem of maximization type can be converted to minimization type by subtracting all the elements of the given profit matrix from the largest element.

## Restrictions on Assignment

Sometimes due to technical or other difficulties do not permit the assignment of a particular facility to a particular job. In such a situation, the difficulty can be overcome by assigning a very high cost (say, infinity i.e. $\infty$) to the corresponding cell.

Because of assigning an infinite penalty to such a cell the activity will be automatically excluded from the optimal solution. Then apply the usual procedure to find the optimal assignment.

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## Assignment as a Minimum Cost Flow Problem

You can use the min cost flow solver to solve special cases of the assignment problem .

In fact, min cost flow can often return a solution faster than either the MIP or CP-SAT solver. However, MIP and CP-SAT can solve a larger class of problems than min cost flow, so in most cases MIP or CP-SAT are the best choices.

The following sections present Python programs that solve the following assignment problems using the min cost flow solver:

- A minimal linear assignment example .
- An assignment problem with teams of workers .

## Linear assignment example

This section show how to solve the example, described in the section Linear Assignment Solver , as a min cost flow problem.

## Import the libraries

The following code imports the required library.

## Declare the solver

The following code creates the minimum cost flow solver.

## Create the data

The flow diagram for the problem consists of the bipartite graph for the cost matrix (see the assignment overview for a slightly different example), with a source and sink added.

The data contains the following four arrays, corresponding to the start nodes, end nodes, capacities, and costs for the problem. The length of each array is the number of arcs in the graph.

To make clear how the data is set up, each array is divided into three sub-arrays:

- The first array corresponds to arcs leading out of the source.
- The second array corresponds to the arcs between workers and tasks. For the costs , this is just the cost matrix (used by the linear assignment solver), flattened into a vector.
- The third array corresponds to the arcs leading into the sink.

The data also includes the vector supplies , which gives the supply at each node.

## How a min cost flow problem represents an assignment problem

How does the min cost flow problem above represent an assignment problem? First, since the capacity of every arc is 1, the supply of 4 at the source forces each of the four arcs leading into the workers to have a flow of 1.

Next, the flow-in-equals-flow-out condition forces the flow out of each worker to be 1. If possible, the solver would direct that flow across the minimum cost arc leading out of each worker. However, the solver cannot direct the flows from two different workers to a single task. If it did, there would be a combined flow of 2 at that task, which couldn't be sent across the single arc with capacity 1 from the task to the sink. This means that the solver can only assign a task to a single worker, as required by the assignment problem.

Finally, the flow-in-equals-flow-out condition forces each task to have an outflow of 1, so each task is performed by some worker.

## Create the graph and constraints

The following code creates the graph and constraints.

## Invoke the solver

The following code invokes the solver and displays the solution.

The solution consists of the arcs between workers and tasks that are assigned a flow of 1 by the solver. (Arcs connected to the source or sink are not part of the solution.)

The program checks each arc to see if it has flow 1, and if so, prints the Tail (start node) and the Head (end node) of the arc, which correspond to a worker and task in the assignment.

## Output of the program

Here is the output of the program.

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 entire program

The entire program is shown below.

## Assignment with teams of workers

This section presents a more general assignment problem. In this problem, six workers are divided into two teams. The problem is to assign four tasks to the workers so that the workload is equally balanced between the teams — that is, so each team performs two of the tasks.

For a MIP solver solution to this problem see Assignment with Teams of Workers .

The following sections describe a program that solves the problem using the min cost flow solver.

The following code creates the data for the program.

The workers correspond to nodes 1 - 6. Team A consists of workers 1, 3, and 5, and team B consists of workers 2, 4, and 6. The tasks are numbered 7 - 10.

There are two new nodes, 11 and 12, between the source and workers. Node 11 is connected to the nodes for team A, and Node 12 is connected to the nodes for team B, with arcs of capacity 1. The graph below shows just the nodes and arcs from the source to the workers.

The key to balancing the workload is that the source 0 is connected to nodes 11 and 12 by arcs of capacity 2. This means that nodes 11 and 12 (and therefore teams A and B) can have a maximum flow of 2. As a result, each team can perform at most two of the tasks.

## Create the constraints

The following shows the output of the program.

Team A is assigned tasks 9 and 10, while team B is assigned tasks 7 and 8.

Note that the min cost flow solver is faster for this problem than the MIP solver , which takes around 0.006 seconds.

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 2023-09-21 UTC.

## Nash Balanced Assignment Problem

- Conference paper
- First Online: 21 November 2022
- Cite this conference paper

- Minh Hieu Nguyen 11 ,
- Mourad Baiou 11 &
- Viet Hung Nguyen 11

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13526))

Included in the following conference series:

- International Symposium on Combinatorial Optimization

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2 Citations

In this paper, we consider a variant of the classic Assignment Problem (AP), called the Balanced Assignment Problem (BAP) [ 2 ]. The BAP seeks to find an assignment solution which has the smallest value of max-min distance : the difference between the maximum assignment cost and the minimum one. However, by minimizing only the max-min distance, the total cost of the BAP solution is neglected and it may lead to an inefficient solution in terms of total cost. Hence, we propose a fair way based on Nash equilibrium [ 1 , 3 , 4 ] to inject the total cost into the objective function of the BAP for finding assignment solutions having a better trade-off between the two objectives: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. For this purpose, we introduce the concept of Nash Fairness (NF) solutions based on the definition of proportional-fair scheduling adapted in the context of the AP: a transfer of utilities between the total cost and the max-min distance is considered to be fair if the percentage increase in the total cost is smaller than the percentage decrease in the max-min distance and vice versa.

We first show the existence of a NF solution for the AP which is exactly the optimal solution minimizing the product of the total cost and the max-min distance. However, finding such a solution may be difficult as it requires to minimize a concave function. The main result of this paper is to show that finding all NF solutions can be done in polynomial time. For that, we propose a Newton-based iterative algorithm converging to NF solutions in polynomial time. It consists in optimizing a sequence of linear combinations of the two objective based on Weighted Sum Method [ 5 ]. Computational results on various instances of the AP are presented and commented.

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## An Efficient Primal-Dual Algorithm for Fair Combinatorial Optimization Problems

## Restricted Max-Min Allocation: Integrality Gap and Approximation Algorithm

Bertsimas, D., Farias, V.F., Trichakis, N.: The price of fairness. Oper. Res. January–February 59 (1), 17–31 (2011)

MathSciNet MATH Google Scholar

Martello, S., Pulleyblank, W.R., Toth, P., De Werra, D.: Balanced optimization problems. Oper. Res. Lett. 3 (5), 275–278 (1984)

Article MathSciNet MATH Google Scholar

Kelly, F.P., Maullo, A.K., Tan, D.K.H.: Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49 (3), 237–252 (1997). https://doi.org/10.1057/palgrave.jors.2600523

Article Google Scholar

Ogryczak, W., Luss, H., Pioro, M., Nace, D., Tomaszewski, A.: Fair optimization and networks: a survey. J. Appl. Math. 2014 , 1–26 (2014)

Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multi. Optim. 41 (6), 853–862 (2010)

Heller, I., Tompkins, C.B.: An extension of a theorem of Dantzig’s. Ann. Math. Stud. (38), 247–254 (1956)

Google Scholar

Kuhn, H.W.: The Hungarian method for assignment problem. Naval Res. Logist. Q. 2 (1–2), 83–97 (1955)

Martello, S.: Most and least uniform spanning trees. Discrete Appl. Math. 15 (2), 181–197 (1986)

Beasley, J.E.: Linear programming on Clay supercomputer. J. Oper. Res. Soc. 41 , 133–139 (1990)

Nguyen, M.H, Baiou, M., Nguyen, V.H., Vo, T.Q.T.: Nash fairness solutions for balanced TSP. In: International Network Optimization Conference (INOC2022) (2022)

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## Author information

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INP Clermont Auvergne, Univ Clermont Auvergne, Mines Saint-Etienne, CNRS, UMR 6158 LIMOS, 1 Rue de la Chebarde, Aubiere Cedex, France

Minh Hieu Nguyen, Mourad Baiou & Viet Hung Nguyen

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## Corresponding author

Correspondence to Viet Hung Nguyen .

## Editor information

Editors and affiliations.

ESSEC Business School of Paris, Cergy Pontoise Cedex, France

Ivana Ljubić

IBM TJ Watson Research Center, Yorktown Heights, NY, USA

Francisco Barahona

Georgia Institute of Technology, Atlanta, GA, USA

Santanu S. Dey

Université Paris-Dauphine, Paris, France

A. Ridha Mahjoub

Proposition 1 . There may be more than one NF solution for the AP.

Let us illustrate this by an instance of the AP having the following cost matrix

By verifying all feasible assignment solutions in this instance, we obtain easily three assignment solutions \((1-1, 2-2, 3-3), (1-2, 2-3, 3-1)\) , \((1-3, 2-2, 3-1)\) and \((1-3, 2-1, 3-2)\) corresponding to 4 NF solutions (280, 36), (320, 32), (340, 30) and (364, 28). Note that \(i-j\) where \(1 \le i,j \le 3\) represents the assignment between worker i and job j in the solution of this instance. \(\square \)

We recall below the proofs of some recent results that we have published in [ 10 ]. They are needed to prove the new results presented in this paper.

Theorem 2 [ 10 ] . \((P^{*},Q^{*}) = {{\,\mathrm{arg\,min}\,}}_{(P,Q) \in \mathcal {S}} PQ\) is a NF solution.

Obviously, there always exists a solution \((P^{*},Q^{*}) \in \mathcal {S}\) such that

Now \(\forall (P',Q') \in \mathcal {S}\) we have \(P'Q' \ge P^{*}Q^{*}\) . Then

The first inequality holds by the Cauchy-Schwarz inequality.

Hence, \((P^{*},Q^{*})\) is a NF solution. \(\square \)

Theorem 3 [ 10 ] . \((P^{*},Q^{*}) \in \mathcal {S}\) is a NF solution if and only if \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) where \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) .

Firstly, let \((P^{*},Q^{*})\) be a NF solution and \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) . We will show that \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) .

Since \((P^{*},Q^{*})\) is a NF solution, we have

Since \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) , we have \(\alpha ^{*}P^{*}+Q^{*} = 2Q^{*}\) .

Dividing two sides of ( 6 ) by \(P^{*} > 0\) we obtain

So we deduce from ( 7 )

Hence, \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) .

Now suppose \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) and \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) , we show that \((P^{*},Q^{*})\) is a NF solution.

If \((P^{*},Q^{*})\) is not a NF solution, there exists a solution \((P',Q') \in \mathcal {S}\) such that

We have then

which contradicts the optimality of \((P^{*},Q^{*})\) . \(\square \)

Lemma 3 [ 10 ] . Let \(\alpha , \alpha ' \in \mathbb {R}_+\) and \((P_{\alpha }, Q_{\alpha })\) , \((P_{\alpha '}, Q_{\alpha '})\) be the optimal solutions of \(\mathcal {P(\alpha )}\) and \(\mathcal {P(\alpha ')}\) respectively, if \(\alpha \le \alpha '\) then \(P_{\alpha } \ge P_{\alpha '}\) and \(Q_{\alpha } \le Q_{\alpha '}\) .

The optimality of \((P_{\alpha }, Q_{\alpha })\) and \((P_{\alpha '}, Q_{\alpha '})\) gives

By adding both sides of ( 8a ) and ( 8b ), we obtain \((\alpha - \alpha ') (P_{\alpha } - P_{\alpha '}) \le 0\) . Since \(\alpha \le \alpha '\) , it follows that \(P_{\alpha } \ge P_{\alpha '}\) .

On the other hand, inequality ( 8a ) implies \(Q_{\alpha '} - Q_{\alpha } \ge \alpha (P_{\alpha } - P_{\alpha '}) \ge 0\) that leads to \(Q_{\alpha } \le Q_{\alpha '}\) . \(\square \)

Lemma 4 [ 10 ] . During the execution of Procedure Find ( \(\alpha _{0})\) in Algorithm 1 , \(\alpha _{i} \in [0,1], \, \forall i \ge 1\) . Moreover, if \(T_{0} \ge 0\) then the sequence \(\{\alpha _i\}\) is non-increasing and \(T_{i} \ge 0, \, \forall i \ge 0\) . Otherwise, if \(T_{0} \le 0\) then the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) .

Since \(P \ge Q \ge 0, \, \forall (P, Q) \in \mathcal {S}\) , it follows that \(\alpha _{i+1} = \frac{Q_i}{P_i} \in [0,1], \, \forall i \ge 0\) .

We first consider \(T_{0} \ge 0\) . We proof \(\alpha _i \ge \alpha _{i+1}, \, \forall i \ge 0\) by induction on i . For \(i = 0\) , we have \(T_{0} = \alpha _{0} P_{0} - Q_{0} = P_{0}(\alpha _{0}-\alpha _{1}) \ge 0\) , it follows that \(\alpha _{0} \ge \alpha _{1}\) . Suppose that our hypothesis is true until \(i = k \ge 0\) , we will prove that it is also true with \(i = k+1\) .

Indeed, we have

The inductive hypothesis gives \(\alpha _k \ge \alpha _{k+1}\) that implies \(P_{k+1} \ge P_k > 0\) and \(Q_{k} \ge Q_{k+1} \ge 0\) according to Lemma 3 . It leads to \(Q_{k}P_{k+1} - P_{k}Q_{k+1} \ge 0\) and then \(\alpha _{k+1} - \alpha _{k+2} \ge 0\) .

Hence, we have \(\alpha _{i} \ge \alpha _{i+1}, \, \forall i \ge 0\) .

Consequently, \(T_{i} = \alpha _{i}P_{i} - Q_{i} = P_{i}(\alpha _{i}-\alpha _{i+1}) \ge 0, \, \forall i \ge 0\) .

Similarly, if \(T_{0} \le 0\) we obtain that the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) . That concludes the proof. \(\square \)

Lemma 5 [ 10 ] . From each \(\alpha _{0} \in [0,1]\) , Procedure Find \((\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in \mathcal {C}_{0}\) satisfying \(\alpha _{k}\) is the unique element \(\in \mathcal {C}_{0}\) between \(\alpha _{0}\) and \(\alpha _{k}\) .

As a consequence of Lemma 4 , Procedure \(\textit{Find}(\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in [0,1], \forall \alpha _{0} \in [0,1]\) .

By the stopping criteria of Procedure Find \((\alpha _{0})\) , when \(T_{k} = \alpha _{k} P_{k} - Q_{k} = 0\) we obtain \(\alpha _{k} \in C_{0}\) and \((P_{k},Q_{k})\) is a NF solution. (Theorem 3 )

If \(T_{0} = 0\) then obviously \(\alpha _{k} = \alpha _{0}\) . We consider \(T_{0} > 0\) and the sequence \(\{\alpha _i\}\) is now non-negative, non-increasing. We will show that \([\alpha _{k},\alpha _{0}] \cap \mathcal {C}_{0} = \alpha _{k}\) .

Suppose that we have \(\alpha \in (\alpha _{k},\alpha _{0}]\) and \(\alpha \in \mathcal {C}_{0}\) corresponding to a NF solution ( P , Q ). Then there exists \(1 \le i \le k\) such that \(\alpha \in (\alpha _{i}, \alpha _{i-1}]\) . Since \(\alpha \le \alpha _{i-1}\) , \(P \ge P_{i-1}\) and \(Q \le Q_{i-1}\) due to Lemma 3 . Thus, we get

By the definitions of \(\alpha \) and \(\alpha _{i}\) , inequality ( 9 ) is equivalent to \(\alpha \le \alpha _{i}\) which leads to a contradiction.

By repeating the same argument for \(T_{0} < 0\) , we also have a contradiction. \(\square \)

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Nguyen, M.H., Baiou, M., Nguyen, V.H. (2022). Nash Balanced Assignment Problem. In: Ljubić, I., Barahona, F., Dey, S.S., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2022. Lecture Notes in Computer Science, vol 13526. Springer, Cham. https://doi.org/10.1007/978-3-031-18530-4_13

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## OPERATIONS RESEARCH

Lesson 9. solution of assignment problem.

Current course

## scipy.optimize.linear_sum_assignment #

Solve the linear sum assignment problem.

The cost matrix of the bipartite graph.

Calculates a maximum weight matching if true.

An array of row indices and one of corresponding column indices giving the optimal assignment. The cost of the assignment can be computed as cost_matrix[row_ind, col_ind].sum() . The row indices will be sorted; in the case of a square cost matrix they will be equal to numpy.arange(cost_matrix.shape[0]) .

for sparse inputs

The linear sum assignment problem [1] is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a ‘worker’) and vertex j of the second set (a ‘job’). The goal is to find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

where, in the case where the matrix X is square, each row is assigned to exactly one column, and each column to exactly one row.

This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

This implementation is a modified Jonker-Volgenant algorithm with no initialization, described in ref. [2] .

Added in version 0.17.0.

https://en.wikipedia.org/wiki/Assignment_problem

DF Crouse. On implementing 2D rectangular assignment algorithms. IEEE Transactions on Aerospace and Electronic Systems , 52(4):1679-1696, August 2016, DOI:10.1109/TAES.2016.140952

## COMMENTS

The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: ... Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as: ... The resulting graph always has a perfect matching of size + ...

1. For the balanced assignment problem, the cost matrix is a square matrix, where the number of rows and columns are equal to the number of workers or tasks. The cost of assigning a worker to a task is placed in the corresponding cell of the matrix. For example, consider a scenario where there are three workers and three tasks.

Cost matrix C = [c ij] where c ij = cost of man i working on job j Variable x ij = 0 or 1. x ij = 1 if man i is assigned to job j and x ij = 0 otherwise Integer program for the assignment problem: ... The constraint matrix of the assignment problem is very special. It will only produce integer solutions.

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.

The assignment cost for dummy cells are always zero. Step 2: Find the Opportunity Cost Table: (a) Locate the smallest element in each row of the given cost table and then subtract that from each element of that row, and ... Since the cost matrix is not a square matrix the problem is unbalanced. We add a dummy job 5 with corresponding entries ...

Jobs with costs of M are disallowed assignments. The problem is to find the minimum cost matching of machines to jobs. Fig 1 Matrix model of the assignment problem. The network model is in shown in Fig.2. It is very similar to the transportatio external flows are all +1 or -1. The only relevant parameter for the assignment model is arc cost

Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero. Step 3. Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row. Step 4.

Step 3. Cover all the zeros of the matrix with the minimum number of horizontal or vertical lines. Step 4. Since the minimal number of lines is 3, an optimal assignment of zeros is possible and we are finished. Since the total cost for this assignment is 0, it must be. Step 3.

M = matchpairs (Cost,costUnmatched) solves the linear assignment problem for the rows and columns of the matrix Cost. Each row is assigned to a column in such a way that the total cost is minimized. costUnmatched specifies the cost per row of not assigning each row, and also the cost per column of not having a row assigned to each column.

Then the problem is to find an assignment so that the total cost of performing all jobs is minimum. (i.e., which job should be assigned to which person with minimum cost). Such problem is called an Assignment Problem (AP). The tabular form of the assignment problem is as follows. The above table is called the n × n cost-matrix, where c i j are ...

Create the data. The flow diagram for the problem consists of the bipartite graph for the cost matrix (see the assignment overview for a slightly different example), with a source and sink added. Note: The numbering of the workers and tasks is slightly different than in the section Linear Assignment Solver, because the min cost flow solver requires all nodes in the graph to be numbered distinctly.

The Assignment Problem (AP) is a fundamental combinatorial optimization problem. It can be formally defined as follows. Given a set n workers, a set of n jobs and a \(n \times n\) cost matrix whose elements are positive representing the assignment of any worker to any job, the AP aims at finding an one-to-one worker-job assignment (i.e., a bipartite perfect matching) that minimizes certain ...

Consider the transportation problem ( Example 1, Section 5.1. with a cost matrix and demand and supply vectors as follows: C = [ 5 7 9 6 6 7 10 5 7 6 8 1], s = [ 120 140 100], a n d d = [ 100 60 80 120]. A minimum cost flow network that models this problem is given in Figure 5.39. Note that each source is a node and each destination is a node.

I have an assignment problem whose solution I have, i.e. I know the job assignments been done, and I know the minimum cost of the entire problem. My question is can I get the cost matrix from these two information, or can i get some approximation of the cost matrix. Please refer to question Finding integer solutions to a multivariable equation.

Rates of Growth Mathematical Modeling of the Problem Given a Cost Matrix C which lists for each "company" i the "cost" of doing "job" j. Solution is a permutation matrix X : all zeros except for one 1 in each row and column Objective is to minimize the total cost An Ideal Cost Matrix All nonnegative entries An possible assignment of ...

It may be noted that the assignment problem is a variation of transportation problem with two characteristics firstly the cost matrix is a square matrix and secondly the optimum solution for the problem would be such that there would be only one assignment in a row or column of the cost matrix. 9.2 Solution of Assignment Problem

This will generate 130 choices of wich at least 20 are some kind of duplicate of some other. You create your cost matrix from this data - the position of the projects number inside the students choice is its priority (0 == 1st, 1 = 2nd, ...): costs = [[0 if pr not in choice else choice.index(pr) + 1.

In conventional assignment problem, cost is always certain. This paper develops an approach to solve the fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the cost matrix of the assignment problem are triangular fuzzy numbers. Its triangular shaped membership function is defined. The ...

Input data in the cost matrix of the linear assignment problem are not always crisp and sometimes in the practical situations is formulated by the grey systems theory approach. In this way, some ...

The problem is to find an assignment with the minimum total cost. There is a question asking to design a greedy algorithm to solve the problem. It also asks if the greedy algorithm always yields an optimal solution and for the performance class of the algorithm. Here is my attempt at designing an algorithm:

Consider a minimization assignment problem with the following 3×3 cost matrix which represents 3 employees and 3 jobs. 25 40 35 40 60 35 20 40 25 ªº «» «» «»¬¼ First: we add an additional row and column for the matrix as follows: Job1 Job2 Job3 Row Sum Emp.1 25 40 35 R 1 = 100 Emp.2 40 60 35 R 2 = 135 Emp.3 20 40 25 R 3 = 85

method for the solution of assignment problem. The well-known Hungarian method developed by Kuhn and published in 1955 [31] is recognized to be the first practical method for solving the assignment problem. Hungarian Method for solving a minimal assignment problem; I. Subtract the minimum element of each row in the cost matrix [c ij

Solve the linear sum assignment problem. Parameters: cost_matrix array. The cost matrix of the bipartite graph. maximize bool (default: False) Calculates a maximum weight matching if true. Returns: row_ind, col_ind array. An array of row indices and one of corresponding column indices giving the optimal assignment.

1. The modifications made by the Hungarian algorithm to the cost matrix are to add/subtract constants from whole rows/columns. Instead of storing the whole matrix, you can store just the row/column deltas (i.e., the potentials) and, when retrieving a matrix element, add the appropriate one of each to the base cost (recomputed as needed).