job assignment problem python

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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.  

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 .

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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.

Introduction

Assignment problem.

Let C be an n by n matrix representing the costs of each of n workers to perform any of n jobs. The assignment problem is to assign jobs to workers in a way that minimizes the total cost. Since each worker can perform only one job and each job can be assigned to only one worker the assignments represent an independent set of the matrix C .

One way to generate the optimal set is to create all permutations of the indexes necessary to traverse the matrix so that no row and column are used more than once. For instance, given this matrix (expressed in Python):

You could use this code to generate the traversal indexes:

After the call to permute(), the results matrix would look like this:

You could then use that index matrix to loop over the original cost matrix and calculate the smallest cost of the combinations:

While this approach works fine for small matrices, it does not scale. It executes in O( n !) time: Calculating the permutations for an n x n matrix requires n ! operations. For a 12x12 matrix, that’s 479,001,600 traversals. Even if you could manage to perform each traversal in just one millisecond, it would still take more than 133 hours to perform the entire traversal. A 20x20 matrix would take 2,432,902,008,176,640,000 operations. At an optimistic millisecond per operation, that’s more than 77 million years.

The Munkres algorithm runs in O( n ^3) time, rather than O( n !). This package provides an implementation of that algorithm.

This version is based on http://csclab.murraystate.edu/~bob.pilgrim/445/munkres.html

This version was written for Python by Brian Clapper from the algorithm at the above web site. (The Algorithm:Munkres Perl version, in CPAN, was clearly adapted from the same web site.)

Construct a Munkres object:

Then use it to compute the lowest cost assignment from a cost matrix. Here’s a sample program:

Running that program produces:

The instantiated Munkres object can be used multiple times on different matrices.

Non-square Cost Matrices

The Munkres algorithm assumes that the cost matrix is square. However, it’s possible to use a rectangular matrix if you first pad it with 0 values to make it square. This module automatically pads rectangular cost matrices to make them square.

• The module operates on a copy of the caller’s matrix, so any padding will not be seen by the caller.
• The cost matrix must be rectangular or square. An irregular matrix will not work.

Calculating Profit, Rather than Cost

The cost matrix is just that: A cost matrix. The Munkres algorithm finds the combination of elements (one from each row and column) that results in the smallest cost. It’s also possible to use the algorithm to maximize profit. To do that, however, you have to convert your profit matrix to a cost matrix. The simplest way to do that is to subtract all elements from a large value. For example:

The munkres module provides a convenience method for creating a cost matrix from a profit matrix. By default, it calculates the maximum profit and subtracts every profit from it to obtain a cost. If, however, you need a more general function, you can provide the conversion function; but the convenience method takes care of the actual creation of the matrix:

So, the above profit-calculation program can be recast as:

Disallowed Assignments

You can also mark assignments in your cost or profit matrix as disallowed. Simply use the munkres.DISALLOWED constant.

Running this program produces:

http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html

• Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly , 2:83-97, 1955.
• Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly , 3: 253-258, 1956.
• Munkres, J. Algorithms for the Assignment and Transportation Problems. Journal of the Society of Industrial and Applied Mathematics , 5(1):32-38, March, 1957.
• http://en.wikipedia.org/wiki/Hungarian_algorithm

Getting and installing munkres

Because munkres is available via PyPI , if you have pip installed on your system, installing munkres is as easy as running this command:

WARNING: As of version 1.1.0, munkres no longer supports Python 2. If you need to use it with Python 2, install an earlier version (e.g., 1.0.12):

Installing from source

You can also install munkres from source. Either download the source (as a zip or tarball) from http://github.com/bmc/munkres/downloads , or make a local read-only clone of the Git repository using one of the following commands:

Once you have a local munkres source directory, change your working directory to the source directory, and type:

To install it somewhere other than the default location (such as in your home directory) type:

Documentation

Consult the API documentation for details. The API documentation is generated from the source code, so you can also just browse the source .

• http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html

This module is released under the Apache Software License, version 2. See the license file for details.

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scipy.optimize.linear_sum_assignment ¶

Solve the linear sum assignment problem.

The linear sum assignment problem 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

s.t. each row is assignment to at most one column, and each column to at most 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.

The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm.

New in version 0.17.0.

• http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html
• Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly , 2:83-97, 1955.
• Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly , 3: 253-258, 1956.
• Munkres, J. Algorithms for the Assignment and Transportation Problems. J. SIAM , 5(1):32-38, March, 1957.
• https://en.wikipedia.org/wiki/Hungarian_algorithm

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Branch and Bound Search with Examples and Implementation in Python

We’ll try to understand one of the heuristic search techniques in this article. The heuristic technique is a criterion for determining which among several alternatives will be the most effective in achieving a particular goal. Branch and bound search is also known as Uniform Cost Search.

What is the branch and bound search algorithm?

Branch and bound is a search algorithm used for combinatory, discrete, and general mathematical optimization problems. It is comparable to backtracking in that it similarly implements a state-space stream to represent the solution to the problem.

However, it is probably more suited to trying to address optimization problems and only minimization problems, not maximization problems. Statistically speaking, a branch and the bound algorithm find the best solution from the entire search space of possibilities for an NP-Hard problem.

How does the branch and bound search work?

In the branch and bound search strategy, a cost function (denoted by g(X)) is generated that, by using a sequence of operators, assigns a cumulative cost to the path from the start node to the current node X. A cheapest price path already discovered is extended at every step of the search space generation process until we reach the goal state.

Branch and bound search is also referred to as a uniform cost search since it expands the least-cost partial path. The actual distance traveled from the beginning to the current node X, for instance, may be represented as g(X) in the traveling salesman problem.

Steps for the algorithm

If g(X) = 1 for all operators, the branch and bound methodology degenerates into a straightforward breadth-first search. Artificial intelligence considers it to be just as detrimental as depth-first and breadth-first. If we add dynamic programming to it, we can make this better by eliminating redundant paths.

We note that the method typically necessitates creating a solution and evaluating its efficacy. Any technique can be used to develop the answer, and heuristics may be used in testing. The following is the basic structure of an algorithm for developing and testing strategies.

Brand and bound search algorithm in action

To understand the concept more clearly, let’s try to implement the 8 puzzle problem using the branch and bound algorithm. The problem description is given below.

A 3 x 3 board with 8 tiles (each tile has a number ranging from 1 to 8) and a single empty space is provided. The goal is to use the vacant space to arrange the numbers on the tiles so that they match the final arrangement. Four neighboring (left, right, above, and below) tiles can be slid into the available area.

For Example

To avoid searching in sub-trees that do not include an answer node, the search for an answer node can frequently be sped up using an approximation of the cost function. However, instead of using the backtracking method, it does a BFS-style search.

Basically, Branch and Bound involve three different kinds of nodes.

• A live node is a generated node whose children have not yet been produced.
• The children of the E-node, a live node, are now being examined. Or to put it another way, an E-node is a node that is currently expanding.
• A created node that is not to be developed or examined further is referred to as a dead node. A dead node has already extended all of its children.

Cost function: In the search tree, each node X has a corresponding cost. The next E-node can be found using the cost function. The E-node with the lowest cost is the next one. The definition of the cost function is

Implementing the Branch and Bound Search algorithm in Python

In this article, we have learned one of the most effective algorithms knowns as a branch and bound search. This search algorithm helps to solve many common problems like the N-Queen problem, 0-1 Knapsack Problem, Traveling salesman problem, etc. The algorithm is bit modified in each case according to the conditions provided in the problem but the basic idea of the searching method remains the same as explained earlier.

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Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)

• Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
• Implementation of Exhaustive Search Algorithm for Set Packing
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• Introduction to Exact Cover Problem and Algorithm X
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Given a 2D array , arr of size N*N where arr[i][j] denotes the cost to complete the j th job by the i th worker. Any worker can be assigned to perform any job. The task is to assign the jobs such that exactly one worker can perform exactly one job in such a way that the total cost of the assignment is minimized.

Input: arr[][] = {{3, 5}, {10, 1}} Output: 4 Explanation: The optimal assignment is to assign job 1 to the 1st worker, job 2 to the 2nd worker. Hence, the optimal cost is 3 + 1 = 4. Input: arr[][] = {{2500, 4000, 3500}, {4000, 6000, 3500}, {2000, 4000, 2500}} Output: 4 Explanation: The optimal assignment is to assign job 2 to the 1st worker, job 3 to the 2nd worker and job 1 to the 3rd worker. Hence, the optimal cost is 4000 + 3500 + 2000 = 9500.

Different approaches to solve this problem are discussed in this article .

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.
• Test for Optimality : If the minimum number of covering lines is N , an optimal assignment is possible. Else if lines are lesser than N , an optimal assignment is not found 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.

Consider an example to understand the approach:

Let the 2D array be: 2500 4000 3500 4000 6000 3500 2000 4000 2500 Step 1: Subtract minimum of every row. 2500, 3500 and 2000 are subtracted from rows 1, 2 and 3 respectively. 0   1500  1000 500  2500   0 0   2000  500 Step 2: Subtract minimum of every column. 0, 1500 and 0 are subtracted from columns 1, 2 and 3 respectively. 0    0   1000 500  1000   0 0   500  500 Step 3: Cover all zeroes with minimum number of horizontal and vertical lines. Step 4: Since we need 3 lines to cover all zeroes, the optimal assignment is found.   2500   4000  3500  4000  6000   3500   2000  4000  2500 So the optimal cost is 4000 + 3500 + 2000 = 9500

For implementing the above algorithm, the idea is to use the max_cost_assignment() function defined in the dlib library . This function is an implementation of the Hungarian algorithm (also known as the Kuhn-Munkres algorithm) which runs in O(N 3 ) time. It solves the optimal assignment problem. 

Below is the implementation of the above approach:

Time Complexity: O(N 3 ) Auxiliary Space: O(N 2 )

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