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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Assignment Problem

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

  • First, prefix name of what this variable represents.
  • Second is the list of all the variables.
  • Third is the lower bound on this variable.
  • Fourth variable is the upper bound.
  • Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

  • Solving Blending Problem in Python using Gurobi
  • Transshipment Problem in Python Using PuLP

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Benchmarking Linear Assignment Problem Solvers

berhane/LAP-solvers

Folders and files, repository files navigation.

The script benchmarks the performance of Python3 linear assignment problem solvers for random cost matrices of different sizes. These solvers are:

  • https://github.com/scipy/scipy/
  • https://github.com/bmc/munkres
  • does not work with Python 3.6 and 3.7
  • https://github.com/Hrldcpr/Hungarian
  • https://github.com/gatagat/lap In addition, these two solvers are added for Python3
  • Please see the blog post here
  • https://github.com/src-d/lapjv
  • Please note that Christioph has also done a benchmark of LAP solvers
  • https://github.com/cheind/py-lapsolver
  • https://github.com/jdmoorman/laptools

They all formally have O(n 3 ) complexity, but their performance differs substantially based on their implementation and the size of the matrix they are trying to solve. The solvers can be classified based on some unique characteristics.

The purpose of this benchmarking exercise is to see which implementation performs best for a given matrix size. My interest is to use this information to improve the performance of Arbalign and expand its use.

The repo contains the following:

  • benchmark-lap-solvers.py - a Python3 script comparing four/six implementations
  • benchmark-lap-solvers-py3.ipynb - a Jupyter notebook comparing four/six implementations. It has been tested using Python 3.6 and 3.7.

It's simple once you have installed the necessary packages.

If you want to add other solvers to the list, it should be easy to figure out what parts to update in the scripts.

Requirements

  • pip3 install numpy
  • conda install numpy
  • pip3 install matplotlib
  • conda install matplotlib
  • pip3 install scipy==1.4
  • conda install scipy
  • pip3 install munkres
  • conda install munkres
  • pip3 install hungarian
  • conda install -c psi4 hungarian
  • pip3 install lap
  • conda install lap
  • pip3 install lapjv
  • pip3 install lapsolver
  • conda install -c loopbio lapsolver
  • pip3 install laptools (Python 3.5+)

The script will produce output similar to what's shown below. Some things to note are:

  • The timings here corresponds to an average of three Python 3.5.6 runs on CentOS 7 machine with 2.4 GHz Intel Xeon Gold 6148 processor and 192GB of RAM
  • The random matrices are filled with floating point numbers ranging from 0 to the size (# of rows or columns) of the matrix. They are generated using numpy: cost_matrix = matrix_size * np.random.random((matrix_size, matrix_size))
  • Data of timing for solving LAP of random cost matrices of sizes 2 min x 2 min to 2 max x 2 max .
  • plot of timing for LAP solving random cost matrices of sizes 2 min x 2 min to 2 max x 2 max , where min and max are limited to smaller numbers for munkres and scipy in the interest of time.

alt text

If requested via the --printcost flag, it will also print the minimum cost for each random cost matrix by each implementation. This test ensures that the methods are making consistent/correct assignments.

  • scipy==1.4 is much faster than previous versions and it is competitive with the other implementations, especially for larger matrices. This is a great development since it probably gets used more than the other implementations by virtue of scipy's popularity.
  • munkres is much slower than hungarian , lapsolver , scipy , lap.lapjv , and lapjv.lapjv for all matrix sizes
  • hungarian performs well for smaller matrices. For anything larger than 256x256, lapsolver , lap.lapjv and lapjv.lapjv are about an order of magnitude faster than hungarian
  • lap.lapjv is am implementation intended to solve dense matrices. Its sparse matrix solver analog named lap.lapmod is more efficient for larger sparse matrices. Both are implemented in the lap module.
  • lapjv.lapjv has the best performance virtually for all matrix sizes.
  • For the purposes of improving Arbalign , hungarian remains a good choice for most molecular systems I'm interested in which don't have more than 100x100 distance matrices the same type to solve. However, if the tool is to be applied to larger molecules such as proteins and DNA, it would be worthwhile to use lapjv.lapjv , lapsolver , lap.lapjv or lap.lapmod
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Matrix with 4 columns and 3 rows

This matrix is a 3x4 (pronounced "three by four") matrix because it has 3 rows and 4 columns.

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Python doesn't have a built-in type for matrices. However, we can treat a list of a list as a matrix. For example:

We can treat this list of a list as a matrix having 2 rows and 3 columns.

Python Matrix Example

Be sure to learn about Python lists before proceed this article.

Let's see how to work with a nested list.

When we run the program, the output will be:

Here are few more examples related to Python matrices using nested lists.

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Using nested lists as a matrix works for simple computational tasks, however, there is a better way of working with matrices in Python using NumPy package.

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NumPy is a package for scientific computing which has support for a powerful N-dimensional array object. Before you can use NumPy, you need to install it. For more info,

  • Visit: How to install NumPy?
  • If you are on Windows, download and install anaconda distribution of Python. It comes with NumPy and other several packages related to data science and machine learning.

Once NumPy is installed, you can import and use it.

NumPy provides multidimensional array of numbers (which is actually an object). Let's take an example:

As you can see, NumPy's array class is called ndarray .

How to create a NumPy array?

There are several ways to create NumPy arrays.

1. Array of integers, floats and complex Numbers

When you run the program, the output will be:

2. Array of zeros and ones

Here, we have specified dtype to 32 bits (4 bytes). Hence, this array can take values from -2 -31 to 2 -31 -1 .

3. Using arange() and shape()

Learn more about other ways of creating a NumPy array .

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Above, we gave you 3 examples: addition of two matrices, multiplication of two matrices and transpose of a matrix. We used nested lists before to write those programs. Let's see how we can do the same task using NumPy array.

Addition of Two Matrices

We use + operator to add corresponding elements of two NumPy matrices.

Multiplication of Two Matrices

To multiply two matrices, we use dot() method. Learn more about how numpy.dot works.

Note: * is used for array multiplication (multiplication of corresponding elements of two arrays) not matrix multiplication.

Transpose of a Matrix

We use numpy.transpose to compute transpose of a matrix.

As you can see, NumPy made our task much easier.

Access matrix elements, rows and columns

Access matrix elements

Similar like lists, we can access matrix elements using index. Let's start with a one-dimensional NumPy array.

Now, let's see how we can access elements of a two-dimensional array (which is basically a matrix).

Access rows of a Matrix

Access columns of a Matrix

If you don't know how this above code works, read slicing of a matrix section of this article.

Slicing of a Matrix

Slicing of a one-dimensional NumPy array is similar to a list. If you don't know how slicing for a list works, visit Understanding Python's slice notation .

Let's take an example:

Now, let's see how we can slice a matrix.

As you can see, using NumPy (instead of nested lists) makes it a lot easier to work with matrices, and we haven't even scratched the basics. We suggest you to explore NumPy package in detail especially if you trying to use Python for data science/analytics.

NumPy Resources you might find helpful:

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  • Zeros and Ones Array
  • Array Using arange() and shape()
  • Multiplication
  • Access elements
  • Access rows
  • Access columns
  • Slicing of Matrix
  • Useful Resources

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scipy.optimize.quadratic_assignment #

Approximates solution to the quadratic assignment problem and the graph matching problem.

Quadratic assignment solves problems of the following form:

where \(\mathcal{P}\) is the set of all permutation matrices, and \(A\) and \(B\) are square matrices.

Graph matching tries to maximize the same objective function. This algorithm can be thought of as finding the alignment of the nodes of two graphs that minimizes the number of induced edge disagreements, or, in the case of weighted graphs, the sum of squared edge weight differences.

Note that the quadratic assignment problem is NP-hard. The results given here are approximations and are not guaranteed to be optimal.

The square matrix \(A\) in the objective function above.

The square matrix \(B\) in the objective function above.

The algorithm used to solve the problem. ‘faq’ (default) and ‘2opt’ are available.

A dictionary of solver options. All solvers support the following:

Maximizes the objective function if True .

Fixes part of the matching. Also known as a “seed” [2] .

Each row of partial_match specifies a pair of matched nodes: node partial_match[i, 0] of A is matched to node partial_match[i, 1] of B . The array has shape (m, 2) , where m is not greater than the number of nodes, \(n\) .

numpy.random.RandomState }, optional

If seed is None (or np.random ), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed . If seed is already a Generator or RandomState instance then that instance is used.

For method-specific options, see show_options('quadratic_assignment') .

OptimizeResult containing the following fields.

Column indices corresponding to the best permutation found of the nodes of B .

The objective value of the solution.

The number of iterations performed during optimization.

The default method ‘faq’ uses the Fast Approximate QAP algorithm [1] ; it typically offers the best combination of speed and accuracy. Method ‘2opt’ can be computationally expensive, but may be a useful alternative, or it can be used to refine the solution returned by another method.

J.T. Vogelstein, J.M. Conroy, V. Lyzinski, L.J. Podrazik, S.G. Kratzer, E.T. Harley, D.E. Fishkind, R.J. Vogelstein, and C.E. Priebe, “Fast approximate quadratic programming for graph matching,” PLOS one, vol. 10, no. 4, p. e0121002, 2015, DOI:10.1371/journal.pone.0121002

D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski, C. Priebe, “Seeded graph matching”, Pattern Recognit. 87 (2019): 203-215, DOI:10.1016/j.patcog.2018.09.014

“2-opt,” Wikipedia. https://en.wikipedia.org/wiki/2-opt

The see the relationship between the returned col_ind and fun , use col_ind to form the best permutation matrix found, then evaluate the objective function \(f(P) = trace(A^T P B P^T )\) .

Alternatively, to avoid constructing the permutation matrix explicitly, directly permute the rows and columns of the distance matrix.

Although not guaranteed in general, quadratic_assignment happens to have found the globally optimal solution.

Here is an example for which the default method, ‘faq’ , does not find the global optimum.

If accuracy is important, consider using ‘2opt’ to refine the solution.

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

hungarian1

  • For each row of the matrix, find the smallest element and subtract it from every element in its row.
  • Do the same (as step 1) for all columns.
  • Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
  • Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
  • Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.
Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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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.

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Hungarian Algorithm Introduction & Python Implementation

How to use hungarian method to resolve the linear assignment problem..

In this article, I will introduce how to use Hungarian Method to resolve the linear assignment problem and provide my personal Python code solution.

So… What is the linear assignment problem?

The linear assignment problem represents the need to maximize the available resources (or minimize the expenditure) with limited resources. For instance, below is a 2D matrix, where each row represents a different supplier, and each column represents the cost of employing them to produce a particular product. Each supplier can only specialize in the production of one of these products. In other words, only one element can be selected for each column and row in the matrix, and the sum of the selected elements must be minimized (minimized cost expense).

The cost of producing different goods by different producers:

Indeed, this is a simple example. By trying out the possible combinations, we can see that the smallest sum is 13, so supplier A supplies Bubble Tea , supplier B supplies milk tea, and supplier C supplies Fruit Tea . However, such attempts do not follow a clear rule and become inefficient when applied to large tasks. Therefore, the next section will introduce step by step the Hungarian algorithm, which can be applied to the linear assignment problem.

Hungarian Algorithm & Python Code Step by Step

In this section, we will show how to use the Hungarian algorithm to solve linear assignment problems and find the minimum combinations in the matrix. Of course, the Hungarian algorithm can also be used to find the maximum combination.

Step 0. Prepare Operations

First, an N by N matrix is generated to be used for the Hungarian algorithm (Here, we use a 5 by 5 square matrix as an example).

The above code randomly generates a 5x5 cost matrix of integers between 0 and 10.

If we want to find the maximum sum, we could do the opposite. The matrix to be solved is regarded as the profit matrix, and the maximum value in the matrix is set as the common price of all goods. The cost matrix is obtained by subtracting the profit matrix from the maximum value. Finally, the cost matrix is substituted into the Hungarian algorithm to obtain the minimized combination and then remapped back to the profit matrix to obtain the maximized sum value and composition result.

The above code randomly generates a 5x5 profit matrix of integers between 0 and 10 and generate a corresponding cost matrix

By following the steps above, you can randomly generate either the cost matrix or the profit matrix. Next, we will move into the introduction of the Hungarian algorithm, and for the sake of illustration, the following sections will be illustrated using the cost matrix shown below. We will use the Hungarian algorithm to solve the linear assignment problem of the cost matrix and find the corresponding minimum sum.

Example cost matrix:

Step 1. Every column and every row subtract its internal minimum

First, every column and every row must subtract its internal minimum. After subtracting the minimum, the cost matrix will look like this.

Cost matrix after step 1:

And the current code is like this:

Step 2.1. Min_zero_row Function Implementation

At first, we need to find the row with the fewest zero elements. So, we can convert the previous matrix to the boolean matrix(0 → True, Others → False).

Transform matrix to boolean matrix:

Corresponding Boolean matrix:

Therefore, we can use the “min_zero_row” function to find the corresponding row.

The row which contains the least 0:

image

Third, mark any 0 elements on the corresponding row and clean up its row and column (converts elements on the Boolean matrix to False). The coordinates of the element are stored in mark_zero.

Hence, the boolean matrix will look like this:

The boolean matrix after the first process. The fourth row has been changed to all False.

The process is repeated several times until the elements in the boolean matrix are all False. The below picture shows the order in which they are marked.

The possible answer composition:

image

Step 2.2. Mark_matrix Function Implementation

After getting Zero_mat from the step 2–1, we can check it and mark the matrix according to certain rules. The whole rule can be broken down into several steps:

  • Mark rows that do not contain marked 0 elements and store row indexes in the non_marked_row
  • Search non_marked_row element, and find out if there are any unmarked 0 elements in the corresponding column
  • Store the column indexes in the marked_cols
  • Compare the column indexes stored in marked_zero and marked_cols
  • If a matching column index exists, the corresponding row_index is saved to non_marked_rows
  • Next, the row indexes that are not in non_marked_row are stored in marked_rows

Finally, the whole mark_matrx function is finished and then returns marked_zero , marked_rows , marked_cols. At this point, we will be able to decide the result based on the return information.

If we use the example cost matrix, the corresponding marked_zero , marked_rows, and marked_cols are as follows:

  • marked_zero : [(3, 2), (0, 4), (1, 1), (2, 0), (4, 3)]
  • marked_rows : [0, 1, 2, 3, 4]
  • marked_cols : []

Step 3. Identify the Result

At this step, if the sum of the lengths of marked_rows and marked_cols is equal to the length of the cost matrix, it means that the solution of the linear assignment problem has been found successfully, and marked_zero stores the solution coordinates. Fortunately, in the example matrix, we find the answer on the first try. Therefore, we can skip to step 5 and calculate the solution.

However, everything is hardly plain sailing. Most of the time, we will not find the solution on the first try, such as the following matrix:

After Step 1 & 2 , the corresponding matrix, marked_rows, and marked_cols are as follows:

image

The sum of the lengths of Marked_Rows and Marked_Cols is 4 (less than 5).

Apparently, the sum of the lengths is less than the length of the matrix. At this time, we need to go into Step 4 to adjust the matrix.

Step 4. Adjust Matrix

In Step 4, we're going to put the matrix after Step 1 into the Adjust_Matrix function . Taking the latter matrix in Step 3 as an example, the matrix to be modified in Adjust_Matrix is:

The whole function can be separated into three steps:

  • Find the minimum value for an element that is not in marked_rows and not in marked_cols . Hence, we can find the minimum value is 1.

image

  • Subtract the elements which not in marked_rows nor marked_cols from the minimum values obtained in the previous step.

image

  • Add the element in marked_rows , which is also in marked_cols , to the minimum value obtained by Step 4–1.

image

Return the adjusted matrix and repeat Step 2 and Step 3 until the conditions satisfy the requirement of entering Step 5.

Step 5. Calculate the Answer

Using the element composition stored in marked_zero , the minimum and maximum values of the linear assignment problem can be calculated.

image

The minimum composition of the assigned matrix and the minimum sum is 18.

image

The maximum composition of the assigned matrix and the maximum sum is 43.

The code of the Answer_Calculator function is as follows:

The complete code is as follows:

Hungarian algorithm - Wikipedia

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Arithmetic Operations in Pandas

A guide on how to use arithmetic operations in Pandas.

Python 3.12: A Game-Changer in Performance and Efficiency

Exploring the Exciting New Additions, Updates, and Changes in Python 3.12

IMAGES

  1. Assignment Problem with Python 01 Basic Matrix Operation

    assignment problem matrix python

  2. Solving Maximization Assignment Problem with Python

    assignment problem matrix python

  3. how to solve matrix problem in python

    assignment problem matrix python

  4. How To Make A Matrix In Python

    assignment problem matrix python

  5. The Matrix Find Algorithm in Python

    assignment problem matrix python

  6. Matrix in Python-Part2 (Operations)

    assignment problem matrix python

VIDEO

  1. Assignment Problem Lec01

  2. Program to output a random matrix in python without numpy

  3. Python

  4. Matrix Assignment || Python || NxtWave CCBP 4.0 || All Questions & Answers || By SP

  5. the problem matrix: 2077

  6. Python Matrix 2

COMMENTS

  1. python

    Since an assignment problem can be posed in the form of a single matrix, I am wondering if NumPy has a function to solve such a matrix. So far I have found none. Maybe one of you guys know if NumPy/SciPy has an assignment-problem-solve function?

  2. scipy.optimize.linear_sum_assignment

    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').

  3. Solving Assignment Problem using Linear Programming in Python

    Solving Assignment Problem using Linear Programming in Python Avinash Navlani February 24, 2022 assignment problem, Linear Programming, Optimization Techniques, PuLP, python Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

  4. The Assignment Problem & Calculating the Minimum Matrix Sum (Python

    The Assignment Problem & Calculating the Minimum Matrix Sum (Python) Ethan Jarrell · Follow Published in HackerNoon.com · 13 min read · Mar 26, 2018 -- Consider the following problem: Due to...

  5. scipy.optimize.linear_sum_assignment

    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. The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm. Notes New in version 0.17.0.

  6. Linear Sum Assignment Solver

    The following sections present a Python program that solves an assignment problem using the linear sum assignment solver. Import the libraries The code that imports the required library is...

  7. The Assignment Problem & Calculating the Minimum Matrix Sum (Python

    March 23rd 2018 13min by @ ethan.jarrell 13,824 reads programming # programming # the-assignment-problem # minimum-matrix-sum # python # software-development 1x Read by Dr. One (en-US) Audio Presented by @ ethan.jarrell Ethan Jarrell Consider the following problem: Due to neglect, your home is in serious need of repair.

  8. ASSIGNMENT PROBLEM (OPERATIONS RESEARCH) USING PYTHON

    The Assignment Problem is a special type of Linear Programming Problem based on the following assumptions: It aims at minimizing the cost or time associated with completing a certain number...

  9. GitHub

    The random matrices are filled with floating point numbers ranging from 0 to the size (# of rows or columns) of the matrix. They are generated using numpy: cost_matrix = matrix_size * np.random.random((matrix_size, matrix_size)) Data of timing for solving LAP of random cost matrices of sizes 2 min x 2 min to 2 max x 2 max.

  10. Python Matrix and Introduction to NumPy

    A matrix is a two-dimensional data structure where numbers are arranged into rows and columns. For example: This matrix is a 3x4 (pronounced "three by four") matrix because it has 3 rows and 4 columns. Python Matrix Python doesn't have a built-in type for matrices. However, we can treat a list of a list as a matrix. For example:

  11. scipy.optimize.quadratic_assignment

    Quadratic assignment solves problems of the following form: min P trace ( A T P B P T) s.t. P ϵ P where P is the set of all permutation matrices, and A and B are square matrices. Graph matching tries to maximize the same objective function.

  12. Hungarian Algorithm for Assignment Problem

    Approach: The idea is to use the Hungarian Algorithm to solve this problem. The algorithm is as follows: For each row of the matrix, find the smallest element and subtract it from every element in its row. Repeat the step 1 for all columns. Cover all zeros in the matrix using the minimum number of horizontal and vertical lines.

  13. Hungarian Algorithm for Assignment Problem

    The Hungarian algorithm, aka Munkres assignment algorithm, utilizes the following theorem for polynomial runtime complexity ( worst case O (n3)) and guaranteed optimality: If a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an op...

  14. Hands-On Linear Programming: Optimization With Python

    In this tutorial, you'll use two Python packages to solve the linear programming problem described above: SciPy is a general-purpose package for scientific computing with Python. PuLP is a Python linear programming API for defining problems and invoking external solvers. SciPy is straightforward to set up.

  15. Solving an Assignment Problem

    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...

  16. Hungarian Algorithm Introduction & Python Implementation

    What is the linear assignment problem? The linear assignment problem represents the need to maximize the available resources (or minimize the expenditure) with limited resources. For instance, below is a 2D matrix, where each row represents a different supplier, and each column represents the cost of employing them to produce a particular product.

  17. python

    Add a comment. 1. This may not be the most efficient way but iteration is passed on to numpy so this may be faster: import numpy as np from scipy.optimize import linear_sum_assignment np.random.seed (0) # define tasks t = np.random.rand (5) # define workers w = np.random.rand (3) W, T = np.meshgrid (w, t) cost_matrix = abs (T-W) Share. Improve ...

  18. python

    I have a problem of assigning 7 possible workers to 3 machines. There is a cost when a worker is assigned to a machine as well as when a worker is idle. ... Construct an assignment matrix - Python. 3. python parallel assignment of 3 variables. 1. Assignment problem with 2 workers per job. 1. How to create an assignment problem cost matrix. 2 ...

  19. Python's Assignment Operator: Write Robust Assignments

    To create a new variable or to update the value of an existing one in Python, you'll use an assignment statement. This statement has the following three components: A left operand, which must be a variable. The assignment operator ( =) A right operand, which can be a concrete value, an object, or an expression.

  20. Construct an assignment matrix

    1. This row: matrix [n, m] = sum (1 for item in b if item== (i)) counts the occurrences of i in b and saves the result to matrix [n, m]. Each cell of the matrix will contain either the number of 1's in b (i.e. 2) or the number of 2's in b (i.e. 2) or the number of 3's in b (i.e. 6). Notice that this value is completely independent of j, which ...

  21. python

    In the assignment, the size of m doesn't change, so I think it will be faster if the assignment is done in place. m = scipy.array([[i * j for j in range(10)] for i in range(10)]) I am worried that in the above code, a temporary matrix is created holding the results, and then m is assigned to this value. This is inefficient because it involves ...