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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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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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Optimizing Job Assignments with Python: A Greedy Approach
In this article, we will learn the skill of job assignment which is a very important topic in the field of Operations Research. For this, we will utilize Python programming language and the Numpy library for the same. We will also solve a small case on a job assignment.
Job assignment involves allocating tasks to workers while minimizing overall completion time or cost. Python’s greedy algorithm, combined with NumPy, can solve such problems by iteratively assigning jobs based on worker skills and constraints, enabling efficient resource management in various industries.
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What is a Job Assignment?
Let us understand what a job assignment is with an example. In our example, three tasks have to be completed. Three workers have different sets of skills and take different amounts of time to complete the above-mentioned tasks. Now our goal is to assign the jobs to the workers to minimize the period of completing the three tasks.
Now, we solve the above problem using the concepts of Linear programming. Now there are certain constraints as well, each worker can be assigned only a single job at a time. Our objective function is the sum of all the time taken by the workers and minimize it. Let us now solve this problem using the power of the Numpy library of Python programming language.
Let us now look at the output of the problem.
From the output, we can see that The assignment is complete and optimized. Let us now look at a small case and understand the job assignment further.
A Real-World Job Assignment Scenario
Continuing with the example of assigning workers some jobs, in this case, a company is looking to get some work done with the help of some freelancers. There are 15 jobs and we have 10 freelancers. We have to assign jobs to workers in such a way, that we minimize the time as well as the cost of the whole operation. Let us now model this in the Python programming language.
This problem is solved using the greedy algorithm. In short, the greedy algorithm selects the most optimal choice available and does not consider what will happen in the future while making this choice. In the above code, we have randomly generated data on freelancer details. Let us now look at the output of the code.
Thus, we complete our agenda of job assignment while minimizing costs as evidenced by the output.
Assigning jobs optimally is crucial for maximizing productivity and minimizing costs in today’s competitive landscape. Python’s powerful libraries like NumPy make it easy to implement greedy algorithms and solve complex job assignment problems, even with larger sets of jobs and workers. How could you adapt this approach to accommodate dynamic changes in job requirements or worker availability?
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Assignment Problem: Meaning, Methods and Variations | Operations Research
After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.
Meaning of Assignment Problem:
An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.
The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.
Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.
Definition of Assignment Problem:
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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.
The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:
Let us explore all approaches for this problem.
Solution 1: Brute Force
We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).
Solution 2: Hungarian Algorithm
The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).
Solution 3: DFS/BFS on state space tree
A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.
Solution 4: Finding Optimal Solution using Branch and Bound
The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.
There are two approaches to calculate the cost function:
- For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
- For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).
In this article, the first approach is followed.
Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A.
Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red).
Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable.
Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14.
Below diagram shows complete search space diagram showing optimal solution path in green.
Complete Algorithm:
Below is the implementation of the above approach:
Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix. Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.
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Assignment Model | Linear Programming Problem (LPP) | Introduction
What is assignment model.
→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;
i) There is only one assignment.
ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).
→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.
For example:
Suppose there are 'n' tasks, which are required to be performed using 'n' resources.
The cost of performing each task by each resource is also known (shown in cells of matrix)
- In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.
How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model.
→ Assignment Model is a special application of Linear Programming (LP).
→ The mathematical formulation for Assignment Model is given below:
→ Let, C i j \text {C}_{ij} C ij denotes the cost of resources 'i' to the task 'j' ; such that
→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.
→ Subjected to constraint;
(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:
(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;
(iii) x i j x_{ij} x ij is '0' or '1'.
Types of Assignment Problem:
(i) balanced assignment problem.
- It consist of a suqare matrix (n x n).
- Number of rows = Number of columns
(ii) Unbalanced Assignment Problem
- It consist of a Non-square matrix.
- Number of rows ≠ \not= = Number of columns
Methods to solve Assignment Model:
(i) integer programming method:.
In assignment problem, either allocation is done to the cell or not.
So this can be formulated using 0 or 1 integer.
While using this method, we will have n x n decision varables, and n+n equalities.
So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.
So this method becomes very lengthy and difficult to solve.
(ii) Transportation Methods:
As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.
In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)
But, here in assignment problems, the matrix is a square matrix (m=n).
So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5
But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.
So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.
So, the method becomes lengthy and time consuming.
(iii) Enumeration Method:
It is a simple trail and error type method.
Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.
For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.
The assignments which gives minimum cost is selected as optimal solution.
But, such trail and error becomes very difficult and lengthy.
If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.
(iv) Hungarian Method:
It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.
It is also know as Reduced matrix method or Flood's technique.
There are two main conditions for applying Hungarian Method:
(1) Square Matrix (n x n). (2) Problem should be of minimization type.
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She’s top of her field, hands down.
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“I would love for it to be my career,” she gushed. “I’d like to do hand modeling for as long as I can.”
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This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver. Example. In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3).
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
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. Summary. In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library.
Job assignments, signalling, and efficiency Michael Waldman* This article analyzes a model in which information about a worker's ability is only directly revealed to the firm employing the worker; other firms, however, use the worker's job as-signment as a signal of ability. Three results recur throughout the analysis.
the model to J jobs and I skills, and Section 6 discusses assignment patterns such as job rotation. Section 7 concludes. 2 Related Literature Although there is a large literature on job assignment, as nicely summarized in Sattinger (1993) and Valsecchi (2000), most papers on the topic assume
The Assignment Problem is a special type of Linear Programming Problem based on the following assumptions: However, solving this task for increasing number of jobs and/or resources calls for…
The general LP assignment model with n workers and n jobs is represented below Minimize: 1 1 n n ij ij i j d x = = ∑∑ Subject to: 1 1 n ij j x = ∑ = i = 1, 2, ….,n (one worker assigned to one job) 1 1 n ij i x = ∑ = j = 1, 2, ….,n (one job assigned only to one worke r) xij =1or 0 for all i and j 5.5.2 The Assignment problem
In the simple job assignment problem, at most one task (job) should be assigned to each employee; this constraint is relaxed in the multiple job assignment problem. ... The power model encompasses ...
Assignment Problem. The assignment problem is a special case of linear programming problem; it is one of the fundamental combinational optimization problems in the branch of optimization or operations research in mathematics. Its goal consists in assigning m resources (usually workers) to n tasks (usually jobs) one a one to one basis while ...
Solution: The task is to assign 1 job to 1 person so that the total number of hours are minimized. So, the first step in the assignment model would be to deduce all the numbers by the smallest number in the row. Hence, the smallest number becomes 0 and then we can target the zeroes to arrive at a conclusion.
bution of earnings. We also analyze an assignment model with variable proportions and find that in the Cobb-Douglas case, a rise in the inequality of ability always narrows the range of earnings. I. Introduction This is a paper about the role of job assignment in the distribution of earnings. The importance of job assignment can be understood from
While there is a large literature on job assignment, as nicely summarized in Sattinger (1993) and Valsecchi (2000), the model discussed in this paper is closest to the papers by Waldman (1984) and Bernhardt (1995), in that it abstracts from strategic considerations of the employees. Both authors assume that the ability of an employee is only ...
Job assignment involves allocating tasks to workers while minimizing overall completion time or cost. Python's greedy algorithm, combined with NumPy, can solve such problems by iteratively assigning jobs based on worker skills and constraints, enabling efficient resource management in various industries. Recommended: Maximizing Cost Savings ...
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 ...
In economic terms, this means that wages at the bottom of the distribution depend most heavily on the sensitivity of output to job assignment in the lower-skilled jobs, and less so in the higher-skilled jobs. Conversely, at the opposite end of the wage profile, we have. 1 1. w(1) = β(1) − Z μ(z)dβ(z)dy. (10) 0 Z z=y.
The most well known quantitative model for assigning workers to jobs is the Assignment Model, which has been utilized for a long time in numerous scheduling applications (Pinedo 1995). The traditional assignment problem seeks to assign a given number of workers to a given set of tasks
We examine the mapping of the distribution of ability onto earnings in a hierarchical job assignment model. Workers are assigned to a continuum of jobs in fixed proportions, ordered by sensitivity to ability. The model implies a novel marginal productivity interpretation of wages. We derive comparative statics for changes in technology and in the distribution of ability. We find conditions ...
Solution Help. Hungarian method calculator. 1. A computer centre has 3expert programmers. The centre wants 3 application programmes to be developed. The head of thecomputer centre, after studying carefully the programmes to be developed, estimates the computer time in minutes required by the experts for the application programmes as follows.
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 ...
Solution 1: Brute Force. We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O (n!). Solution 2: Hungarian Algorithm. The optimal assignment can be found using the Hungarian algorithm.
Jobs with costs of M are disallowed assignments. The problem is to find the minimum cost matching of machines to jobs. Figure 12. Matrix model of the assignment problem. The network model is in Fig. 13. It is very similar to the transportation model except the external flows are all +1 or -1.
There are two main conditions for applying Hungarian Method: (1) Square Matrix (n x n). (2) Problem should be of minimization type. Assignment model is a special application of Linear Programming Problem (LPP), in which the main objective is to assign the work or task to a group of individuals such that; i) There is only one assignment.
1 Introduction. Due to their effectiveness in replicating unemployment, worker turnover, and wage dynamics, on-the-job search models are extensively used to evaluate labor market policies. 1 Embedded in the standard job ladder model is a single friction, modeled as a Poisson process, that prevents the reallocation of workers into more productive jobs. . In this paper, we extend the standard ...
Specifically, the final rule defines the term "senior executive" to refer to workers earning more than $151,164 annually who are in a "policy-making position.". The FTC estimates that banning noncompetes will result in: Reduced health care costs: $74-$194 billion in reduced spending on physician services over the next decade.
Job Summary: Supervises, guides, and/or instructs the work assignments of subordinate staff. Supervises at least one function, such as job development and analysis, compensation, recruitment, benefits analysis, exam development, employee relations, and/or policy development. Supervises all activities related to area of expertise.
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Attacked Supervisor. 228. 4/24/2024 4:59 PM PT. Getty Compsosite. Kamala Harris ' Secret Service team faced a violent threat this week, having to defend one of their agents ... from one of their ...