indicate a method of solving an assignment problem

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:

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Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Related Articles:

• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
• Linear Programming: Applications, Definitions and Problems

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Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

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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 .

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

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Linear assignment problem: Understanding the core of assignment method

1. introduction to the linear assignment problem, 2. the basics of assignment method, 3. formulating the linear assignment problem, 4. solving the linear assignment problem using hungarian algorithm, 5. understanding the optimality conditions in assignment method, 6. applications of linear assignment problem in real life, 7. extensions and variations of the assignment method, 8. challenges and limitations of the linear assignment problem, 9. harnessing the power of assignment method for optimization.

The Linear Assignment Problem is a fundamental concept in the field of optimization. It is a mathematical formulation used to solve the problem of assigning a set of jobs to a set of workers, where each worker is capable of doing only one job at a time. This problem arises in many real-world applications , such as scheduling tasks in a manufacturing plant, assigning students to courses, and routing vehicles to destinations. The goal of the Linear Assignment Problem is to find the optimal assignment that satisfies all constraints and minimizes the total cost of the assignments.

Here are some in-depth insights into the Linear Assignment Problem:

1. The Linear Assignment Problem can be represented using a matrix. The rows of the matrix represent the workers, and the columns represent the jobs. Each cell in the matrix represents the cost of assigning a particular job to a particular worker. The problem is to find a set of assignments that minimizes the total cost.

2. The Hungarian Algorithm is an efficient algorithm used to solve the Linear Assignment Problem. It is based on the principle of reducing the problem to a series of smaller sub-problems, each of which can be solved easily. The algorithm has a time complexity of O(n^3), where n is the number of workers or jobs.

3. The Linear Assignment Problem can be extended to the case where each worker can do more than one job. This is known as the Generalized Assignment Problem. In this case, each worker has a capacity, and each job has a requirement. The goal is to assign the jobs to the workers in a way that satisfies the capacity and requirement constraints and minimizes the total cost.

4. The Linear Assignment Problem has many practical applications. For example, in the field of computer vision, it is used to solve the problem of object recognition. In this case, each object is represented by a set of features, and each feature is assigned a weight based on its importance. The goal is to assign the features to the objects in a way that maximizes the total weight.

The Linear Assignment Problem is a powerful tool for solving the problem of assigning jobs to workers. It has many practical applications and can be solved efficiently using the Hungarian Algorithm.

Introduction to the Linear Assignment Problem - Linear assignment problem: Understanding the core of assignment method

In this section, we will delve into the fundamentals of the assignment method, a powerful tool used to solve linear assignment problems. Understanding the core concepts behind this method is crucial for effectively tackling such problems and optimizing resource allocation . From different perspectives, let's explore the intricacies of the assignment method and how it can be applied in various scenarios.

1. Definition of Assignment Method:

At its core, the assignment method is a mathematical technique used to determine the optimal assignment of a set of tasks to a set of resources. It aims to minimize the total cost or maximize the total benefit of the assigned tasks. The method assigns each task to a single resource, ensuring that all tasks are completed and each resource is utilized optimally.

2. Formulating the Assignment Problem:

The assignment problem can be represented as a matrix, where the rows represent the tasks and the columns represent the resources. Each element in the matrix represents the cost or benefit associated with assigning a particular task to a specific resource. The goal is to find the assignment that minimizes the total cost or maximizes the total benefit.

For example, consider a scenario where a company needs to assign four employees (A, B, C, D) to four projects (X, Y, Z, W). The matrix would represent the costs or benefits associated with each assignment, such as A-X, B-Y, C-Z, and D-W.

3. Solution Approaches:

The assignment problem can be solved using different approaches, including the Hungarian method, the auction algorithm, or the shortest path algorithm. Each method has its own advantages and is suitable for specific problem types.

4. The Hungarian Method:

The Hungarian method is one of the most commonly used techniques to solve the assignment problem. It involves finding a series of augmenting paths in the assignment matrix until an optimal assignment is achieved. This method guarantees an optimal solution and has a time complexity of O(n^3), making it efficient for small to medium-sized problems.

For instance, using the Hungarian method, we can find the optimal assignment for the previously mentioned scenario. By iteratively identifying augmenting paths and adjusting the assignment matrix, we can determine the best employee-project assignments that minimize the overall cost or maximize the total benefit.

5. Applications of the Assignment Method:

The assignment method finds applications in various fields, such as project management, workforce scheduling, transportation logistics, and resource allocation. It allows organizations to optimize their operations by efficiently assigning tasks to available resources, minimizing costs, and maximizing productivity.

For example, a courier company can utilize the assignment method to determine the most efficient routes for its delivery drivers, considering factors like distance, traffic conditions, and delivery deadlines.

In summary, the assignment method is a powerful mathematical technique used to solve linear assignment problems. By formulating the problem as a matrix and applying various solution approaches like the Hungarian method, organizations can optimize their resource allocation and maximize efficiency. The versatility of the assignment method makes it applicable in diverse fields, ensuring optimal task assignments and improved overall performance.

The Basics of Assignment Method - Linear assignment problem: Understanding the core of assignment method

When it comes to solving real-world optimization problems, the linear assignment problem (LAP) is a fundamental concept that plays a crucial role in various fields such as operations research, computer science, and economics. The LAP involves assigning a set of tasks to a set of agents in the most efficient manner possible, taking into consideration certain constraints and objectives. This problem can be represented mathematically as a bipartite graph, where one set of vertices represents the tasks and the other set represents the agents. Each edge between these sets represents the cost or benefit associated with assigning a particular task to a specific agent.

From an operational perspective, formulating the LAP requires careful consideration of several factors. Here are some key insights from different points of view:

1. Objective Function:

- The objective function defines what we aim to optimize in the LAP. It could be minimizing costs, maximizing benefits, or achieving a balance between the two.

- For example, consider a scenario where a company needs to assign delivery routes to its drivers. The objective might be to minimize the total distance traveled by all drivers.

2. Constraints:

- Constraints impose limitations on the assignment process. These can include restrictions on task-agent compatibility, capacity constraints for agents, or exclusivity requirements.

- For instance, if we have a group of students who need to be assigned to different projects based on their skills and preferences, we must ensure that each student is assigned to only one project.

3. cost or Benefit matrix :

- To formulate the LAP mathematically, we need to construct a cost or benefit matrix that captures the relationship between tasks and agents.

- In our previous example of delivery route assignments, this matrix would contain distances between different locations and drivers.

4. Decision Variables:

- Decision variables represent the assignment decisions made during optimization. They indicate whether a task is assigned to an agent or not.

- In the delivery route assignment scenario, each decision variable could represent whether a driver is assigned to a specific route or not.

5. Mathematical Model:

- By combining the objective function, constraints, cost/benefit matrix, and decision variables, we can create a mathematical model that represents the LAP.

- This model can then be solved using various optimization techniques such as the Hungarian algorithm or linear programming.

In summary, formulating the Linear Assignment Problem involves defining an objective function, considering relevant constraints, constructing a cost or benefit matrix, determining decision variables, and creating

Formulating the Linear Assignment Problem - Linear assignment problem: Understanding the core of assignment method

The Hungarian algorithm is a combinatorial optimization technique that is commonly used to solve the linear assignment problem. It was first introduced by two Hungarian mathematicians, Kuhn and Munkres, in 1955. The algorithm is known for its speed and efficiency in solving large-scale assignment problems. It works by iteratively improving the assignment until an optimal solution is found. The algorithm has been extensively studied and applied in various fields, such as computer science, operations research, and engineering.

Here are some key insights into how the Hungarian algorithm works:

1. The Hungarian algorithm starts by creating a matrix of costs, where each row represents a worker and each column represents a task. The cost of assigning each worker to each task is listed in the matrix.

2. The algorithm then uses a series of steps to find the optimal assignment. It starts by finding the smallest element in each row and subtracting that element from every element in that row. It then finds the smallest element in each column and subtracts that element from every element in that column.

3. The algorithm then identifies the smallest uncovered element in the matrix and assigns the corresponding worker to the corresponding task. If there is a tie for the smallest uncovered element, the algorithm randomly chooses one of the tied elements to assign.

4. The algorithm continues to iteratively improve the assignment by repeating the steps above until an optimal solution is found.

5. One important aspect of the Hungarian algorithm is that it guarantees to find the optimal solution in a finite number of steps. This is because the algorithm reduces the number of uncovered elements in the matrix by at least one in each iteration.

6. The Hungarian algorithm can also handle cases where the number of workers is not equal to the number of tasks. In these cases, the algorithm adds dummy tasks or workers to the matrix to make it square.

7. Let's consider an example to illustrate how the Hungarian algorithm works. Suppose we have three workers and three tasks, and the costs of assigning each worker to each task are given in the following matrix:

Task 1 Task 2 Task 3

The algorithm starts by subtracting the smallest element in each row from every element in that row, and the smallest element in each column from every element in that column. The resulting matrix is:

The algorithm then assigns W1 to Task 1, W2 to Task 3, and W3 to Task 2, resulting in a total cost of 5+0+4=9. This is the optimal solution to the assignment problem.

Overall, the Hungarian algorithm provides an efficient and effective method for solving the linear assignment problem. Its ability to handle large-scale problems and guarantee an optimal solution makes it a valuable tool for a wide range of applications .

Solving the Linear Assignment Problem using Hungarian Algorithm - Linear assignment problem: Understanding the core of assignment method

Understanding the optimality conditions in the assignment method is crucial for solving the linear assignment problem effectively. By comprehending these conditions, we can gain valuable insights into how the assignment method works and why it produces optimal solutions. In this section, we will delve into the optimality conditions from various perspectives, exploring their significance and implications.

1. Objective Function: The objective function in the assignment method aims to minimize or maximize a certain criterion, such as cost or profit. For example, in a transportation problem where the goal is to minimize total transportation costs, the objective function would be to minimize the sum of costs associated with assigned tasks. Understanding this condition helps us grasp the underlying purpose of the assignment method and its alignment with specific optimization goals.

2. Feasibility: Feasibility refers to satisfying all constraints imposed by the problem at hand. In the context of the assignment method, feasibility entails ensuring that each task is assigned to exactly one agent or resource, and each agent or resource is assigned to exactly one task. Violating this condition would result in an infeasible solution. For instance, if two agents are assigned to the same task, it would violate feasibility. Recognizing this condition aids in identifying potential errors or inconsistencies in assignments.

3. Optimality: The optimality condition determines when a solution obtained through the assignment method is considered optimal. It states that an assignment is optimal if there is no other feasible assignment that yields a better objective function value. In other words, no reassignment can improve the overall criterion being optimized. This condition ensures that we have found the best possible solution within the given constraints and objectives.

4. Complementary Slackness: Complementary slackness is a fundamental concept in linear programming and plays a significant role in understanding optimality conditions in the assignment method. It states that for every pair of decision variables (assignment variables) and constraint equations (feasibility constraints), either both are zero or both have positive values. This condition implies that if a task is assigned to an agent, the corresponding constraint equation must be satisfied, and vice versa. By considering complementary slackness, we can verify the optimality of assignments and ensure that no unutilized resources or unassigned tasks exist.

To illustrate these optimality conditions, let's consider a scenario where a company needs to assign four employees (A, B, C, D) to four different projects (P1, P2, P3, P4). The objective is to minimize the total cost associated with each assignment.

Understanding the Optimality Conditions in Assignment Method - Linear assignment problem: Understanding the core of assignment method

The linear assignment problem, a fundamental concept in optimization theory, finds its applications in various real-life scenarios . From resource allocation to scheduling and matching problems, the linear assignment problem offers a powerful framework for solving complex decision-making tasks efficiently. By understanding the core principles of the assignment method, we can gain valuable insights into how this mathematical technique is applied in different fields.

1. Resource Allocation: One of the most common applications of the linear assignment problem is in resource allocation. For example, in transportation logistics, it can be used to optimize the assignment of vehicles to delivery routes, ensuring efficient utilization of resources while minimizing costs. Similarly, in project management, it can help allocate tasks to team members based on their skills and availability, maximizing productivity.

2. Workforce Scheduling: The linear assignment problem also finds extensive use in employee scheduling. For instance, in healthcare settings such as hospitals or clinics, it can be employed to assign nurses or doctors to shifts based on their expertise and workload requirements. By optimizing these assignments, organizations can ensure adequate coverage while minimizing overtime and maintaining employee satisfaction .

3. Matching Problems: Another area where the linear assignment problem proves invaluable is in matching problems. This includes applications like matching medical students to residency programs or assigning students to schools based on their preferences and qualifications. By formulating these problems as linear assignment problems, optimal matches can be determined efficiently, taking into account various constraints and preferences.

4. Facility Location: The linear assignment problem can also aid in determining optimal facility locations. For instance, in retail planning, it can be used to assign customers to nearby stores based on factors like distance and demand patterns. By solving this optimization problem, businesses can strategically position their facilities to maximize customer reach and minimize transportation costs .

5. Data Association: In computer vision and pattern recognition tasks such as object tracking or image registration, the linear assignment problem plays a crucial role in data association. It helps establish correspondences between observed data and known models, enabling accurate tracking or alignment. By solving the assignment problem, the most likely associations can be determined, leading to improved performance in these applications.

6. Network Flow Optimization: The linear assignment problem can also be applied to optimize network flows. For example, in telecommunications, it can help allocate bandwidth to different users or routes based on their demands and available resources. By solving this optimization problem, network operators can ensure efficient utilization of network capacity while meeting user requirements.

These examples highlight the versatility and practicality of the linear assignment problem in real-life scenarios. By

Applications of Linear Assignment Problem in Real Life - Linear assignment problem: Understanding the core of assignment method

When it comes to the assignment method, there are several extensions and variations that can be applied. These extensions and variations add more complexity and flexibility to the method, allowing it to solve more complicated problems. They can also provide additional insights into the problem and help to optimize the assignment process. From different point of views, these extensions and variations can be used to model a wide range of scenarios, such as the assignment of personnel to tasks, the allocation of resources to projects, and the assignment of frequencies to radio channels. Some of the most common extensions and variations of the assignment method are listed below.

1. Multiple Assignments: This extension allows more than one assignment to be made to each agent or task. For example, a teacher may be assigned to teach multiple classes, or a delivery truck may be assigned to make multiple deliveries.

2. Partial Assignments: This variation allows for partial assignments to be made, where only a portion of the agent or task's capacity is utilized. For instance, an employee may only work part-time, or a machine may only be used for a certain number of hours per day.

3. Cost Matrices with Unequal Elements: This variation allows for the cost matrix to have unequal elements, where the cost of assigning one agent to a task may be different from the cost of assigning another agent to the same task. This can be particularly useful when dealing with different skill levels or availability of agents.

4. Non-Rectangular Cost Matrices: This variation allows for the cost matrix to have a non-rectangular shape, where the number of agents and tasks are not equal. For example, in a transportation problem, the number of supply and demand points may be different.

5. Multiple Objectives: This extension allows for multiple objectives to be considered simultaneously, such as minimizing cost and maximizing efficiency. This can be achieved through the use of multi-objective optimization techniques.

The extensions and variations of the assignment method provide a powerful tool for solving complex assignment problems in a wide range of scenarios. By adding more complexity and flexibility to the method, these extensions and variations can help to optimize the assignment process and provide additional insights into the problem.

Extensions and Variations of the Assignment Method - Linear assignment problem: Understanding the core of assignment method

The linear assignment problem is a fundamental concept in optimization theory that involves assigning a set of resources to a set of tasks, with the objective of minimizing the total cost or maximizing the total benefit. While the assignment method provides an efficient solution for many real-world problems , it also comes with its own set of challenges and limitations. Understanding these challenges is crucial for effectively applying the assignment method and obtaining optimal solutions.

1. Complexity: The linear assignment problem can become computationally complex as the number of resources and tasks increases. The time required to solve the problem grows exponentially, making it impractical for large-scale applications. For example, consider a scenario where there are 100 resources and 100 tasks. The number of possible assignments to evaluate would be 100 factorial (100!), which is an astronomically large number.

2. Non-linearity: Despite its name, the linear assignment problem can involve non-linear relationships between resources and tasks. This occurs when the cost or benefit associated with an assignment depends on factors other than just the resource-task pair itself. For instance, if assigning a particular resource to a task affects the performance of other resources or tasks, the problem becomes more complex and may require additional considerations.

3. Limited flexibility: The assignment method assumes that each resource can only be assigned to one task and vice versa. However, in some scenarios, allowing multiple assignments or partial assignments may lead to better overall outcomes. For example, in workforce scheduling, it may be beneficial to assign multiple employees to a single task to ensure timely completion or handle unexpected events.

4. Uncertainty and dynamic environments: The linear assignment problem assumes that all information regarding costs, benefits, and resource/task characteristics is known with certainty at the time of assignment. However, in real-world situations, such information may be uncertain or subject to change over time. This introduces additional complexity as decisions need to be made under uncertainty or adaptively in dynamic environments.

5. Lack of consideration for qualitative factors: The assignment method primarily focuses on quantitative factors such as costs or benefits. It may not adequately consider qualitative factors, such as skill levels, preferences, or compatibility between resources and tasks. Ignoring these qualitative aspects can lead to suboptimal assignments that do not fully utilize the capabilities or preferences of resources.

6. Scalability: As the number of resources and tasks increases, the assignment problem becomes more challenging to solve optimally. While approximation algorithms exist to handle large-scale problems, they may sacrifice optimality for computational efficiency. Balancing scalability and solution

Challenges and Limitations of the Linear Assignment Problem - Linear assignment problem: Understanding the core of assignment method

The assignment method is a powerful tool for solving optimization problems , particularly the linear assignment problem. Throughout this blog, we have explored the core concepts and techniques behind the assignment method, shedding light on its potential applications and benefits. In this concluding section, we will delve deeper into the significance of harnessing the power of the assignment method for optimization.

1. Versatility: One of the key advantages of the assignment method is its versatility in tackling a wide range of optimization problems. Whether it is assigning tasks to workers, matching students to schools, or allocating resources to projects, the assignment method can be applied to various scenarios. By formulating these problems as linear assignment problems, we can leverage the efficiency and effectiveness of the assignment method to find optimal solutions.

For example, consider a company that needs to assign a set of tasks to its employees based on their skills and availability. By using the assignment method, the company can optimize task assignments by considering factors such as skill compatibility and workload distribution. This not only ensures efficient resource utilization but also enhances overall productivity.

2. Complexity Reduction: The assignment method simplifies complex optimization problems by transforming them into linear assignment problems. This reduction in complexity allows us to apply well-established algorithms and techniques specifically designed for linear assignment problems. As a result, we can efficiently solve large-scale optimization problems that would otherwise be computationally challenging or time-consuming.

For instance, imagine a logistics company that needs to determine the most cost-effective routes for delivering goods to multiple destinations while considering factors like distance, traffic conditions, and delivery deadlines. By formulating this problem as a linear assignment problem using the assignment method , the company can quickly find optimal routes that minimize costs and maximize customer satisfaction.

3. Optimality Guarantee: The assignment method guarantees finding an optimal solution for linear assignment problems. This means that once we have formulated our problem correctly and applied the appropriate algorithm, we can be confident that the solution obtained is indeed the best possible solution. This optimality guarantee is particularly valuable in situations where making suboptimal decisions can have significant consequences.

For example, in healthcare resource allocation, the assignment method can be used to optimize the assignment of medical staff to different departments or shifts. By considering factors such as expertise, workload, and patient needs, the assignment method ensures that the allocation is optimal, leading to improved patient care and overall operational efficiency.

4. Scalability: The assignment method exhibits excellent scalability, allowing us to handle optimization problems with a large number of variables and constraints. This scalability is crucial in

Harnessing the Power of Assignment Method for Optimization - Linear assignment problem: Understanding the core of assignment method

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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.

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.

Suggested Notes:

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Indirect cost less than Crash Cost

Basics of Program Evaluation and Review Technique (PERT)

Numerical on PERT (Program Evaluation and Review Technique)

Network Analysis - Dealing with Network Construction Basics

Construct a project network with predecessor relationship | Operation Research | Numerical

Graphical Method | Methods to solve LPP | Linear Programming

Basics of Linear Programming

Linear Programming Problem (LPP) Formulation with Numericals

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

• Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)
• Introduction to Exact Cover Problem and Algorithm X
• Greedy Approximate Algorithm for Set Cover Problem
• Job Assignment Problem using Branch And Bound
• Implementation of Exhaustive Search Algorithm for Set Packing
• Channel Assignment Problem
• Chocolate Distribution Problem | Set 2
• Transportation Problem | Set 1 (Introduction)
• OLA Interview Experience | Set 11 ( For Internship)
• Top 20 Greedy Algorithms Interview Questions
• Job Sequencing Problem - Loss Minimization
• Prim's Algorithm (Simple Implementation for Adjacency Matrix Representation)
• Data Structures and Algorithms | Set 21
• Algorithms Quiz | Sudo Placement : Set 1 | Question 4
• Adobe Interview Experience | Set 55 (On-Campus Full Time for MTS profile)
• Amazon Interview Experience | Set 211 (On-Campus for Internship)
• OYO Rooms Interview Experience | Set 3 (For SDE-II, Gurgaon)
• C# Program for Dijkstra's shortest path algorithm | Greedy Algo-7
• Algorithms | Dynamic Programming | Question 7

• 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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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

7 Most Effective Ways For How To Solve Assignment Problems

Here in this blog, CodeAvail experts will explain to you 7 most effective ways for how to solve assignment problems step by step in detail.

How To Solve Assignment Problems

Table of Contents

The most common question asked by many students is how to solve assignment problems. There are many students who are having a hard time with their assignments thats why they are looking for proper guidelines to solve those problems. The assignment is one of the most critical parts of the students’ academic journey, and one always states that I have a lot of assignments to do. Besides this, there is not a single student in the world who has never done any assignment in their academic life. But the students always want to find the best and most effective ways for how to solve assignment problems . In this blog, we have given all the necessary information to help you solve your assignment problems.

How To Solve Assignment Problems-Stepwise Guide

To solve assignment problems, you need to understand the question carefully first. Because if you don’t understand the question, you will not know the information you need to complete your assignments. Follow these steps to know How To Solve assignment Problems :

1) Proper Planning Is Required 

This is the first step when you start solving your assignment, you’ll most likely jump directly into the main thing. The primary thing you will do is pull out of your beg, at that point work your way through the remainder of your assignment. There’s a superior way. Know how much time you need to do an assignment. At that point list down all the various chores that you need to do. Check to what extent it will take you to finish every task to check whether you have to permit yourself additional time. Be practical. When your note down everything then the next step is to find the best place for work.

2) Collect All the required Data Before You Start  

This is the biggest problem of many students, they don’t really follow this step. Make sure you have all the required information to solve assignment problems. Don’t start your assignment without the proper information. Because after starting your assignment when you come back it may be hard to get once again and write with the same flow. It will just destroy your assignment writing flow. If you have planned efficiently, you should know what exactly you want to complete your assignment and set up everything in your study table you’ll require.

3) Set A Timetable For Certain Assignment

Set a proper timetable to achieve each part of your assignment based on how long you think each part of your assignment will take and how much time you have. Give yourself enough time to complete each part and do other nightly routines. Set a proper time and be honest with it. The less time you waste on checking your laptop or mobile, the more quickly you can complete your assignment. If you believe you can complete everything in a half-hour, set a timer and work honestly to complete it.

4) Stay Away From Distractions

You need a quiet place when you are working on your assignment problems. Problems like Maths assignment problems, Computer science problems needs a quiet place to complete. Keep your telephone away from you, stay away from your PC, and make your surroundings as peaceful as possible. Because to solve assignment problems it requires a lot of focus. Giving work your full focus will really make it simpler, in light of the fact that your brain won’t balance various tasks simultaneously. 

Usually, students will attempt to perform multiple tasks, sitting in front of the TV or tuning in to the radio or proceeding to visit on Facebook or Instagram while additionally attempting to do assignments. It will be a lot more enjoyable to do those things after you are done completing your assignment.

5) Taking Breaks is Necessary

If your professor assigns you a lot of assignments to complete, then You want to work straight through hours and hours of assignment if you have a lot to do. But it would probably end up slowing you down and prolonging the whole session.

Get the work done in short periods. Go hard on an assignment, then take a short break to stretch and walk. To keep going, it will re-energize your mind and your body. This strategy will help you solve your assignment problems quickly and help you maintain your assignment’s quality. Try to do your assignment for 1 hour and then take a 10 minutes break.

6) Isolate Yourself

It is one of the best ways to do assignment problems. Because when we isolate ourselves from the outside world, we can do a better concentration on our assignment. The reason is, many outside elements distract our mind from our assignment. When we are going to do our assignment, we should isolate ourselves from our family, social media, and other social activities. 

7) Take Help From Online Service Providers

There are many Excel problems that are hard to solve at that time you can take help from online service providers. The reason for taking help from these online service providers because they have years of experience in their respective field. Plus if you need any help you can contact them because they are always available to deal with your problems. They are well aware of the guidelines provided by universities and colleges and know how to solve assignment problems .

Follow all the steps we mentioned above will help in solving your assignment problems. In this article, we have given all the required information that will help you find the answer on how to solve assignment problems . The assignment problems like Maths and Computer science required a lot of focus and proper time management. Time is everything if you manage your time effectively then you will not find any problem solving your assignment problems before the allotted time. Just put away the thing which you think can distract your mind. And give yourself a break after you complete every task because if you give your rain rest then it will be helpful for you to focus on your next task.

If you are facing a problem with completing your assignment or any other assignment, you can take help from us. Our Assignment providers have helped students across the world in completing their assignments on time, get good grades, and at the same time. There’s no doubt that many teachers are handing out assignments that are hard to complete on time. Some assignments can be about an unknown subject. But the truth is, often, the students need assignment help. It is your best decision when you have a lot of tasks to complete but still want to have free time.

As a result, Our computer science assignment help and programming assignment help experts are available for you to 24*7.

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Gaining insight into crew rostering instances through ML-based sequential assignment

• Original Paper
• Published: 25 June 2024

Cite this article

• Philippe Racette   ORCID: orcid.org/0009-0000-7422-526X 1 , 3 ,
• Frédéric Quesnel 2 ,
• Andrea Lodi 3 , 4 &
• François Soumis 1  

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Crew scheduling is typically performed in two stages. First, solving the crew pairing problem generates sequences of flights called pairings. Then, the pairings are assigned to crew members to provide each person with a full schedule. A common way to do this is to solve an optimization problem called the crew rostering problem (CRP). However, before solving the CRP, the problem instance must be parameterized appropriately while taking different factors such as preassigned days off, crew training, sick leave, reserve duty, or unusual events into account. In this paper, we present a new method for the parameterization of CRP instances for pilots by scheduling planners. A machine learning-based sequential assignment procedure ( seqAsg ) whose arc weights are computed using a policy over state–action pairs for pilots is implemented to generate very fast solutions. We establish a relationship between the quality of the solutions generated by seqAsg and that of solutions produced by a state-of-the-art solver. Based on those results, we formulate recommendations for instance parameterization. Given that the seqAsg procedure takes only a few seconds to run, this allows scheduling workers to reparameterize crew rostering instances many times over the course of the planning process as needed.

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Data availability

The original data used for the experiments shown in this paper can be downloaded directly at the following URL: https://www.gerad.ca/en/papers/G-2014-22/ . Since GENCOL is proprietary software, the code for the related experiments is not available. However, we can provide code to build rosters with sequential assignment upon request.

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Acknowledgements

This work was supported financially by the Natural Sciences and Engineering Research Council of Canada and IBS under Grant no. CRDPJ-477127-14. The authors are grateful for this support.

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Appendix A: Pseudo-code for the feasibility heuristic

In this section, we detail further the feasibility heuristic presented in Sect.  5.2 . We give the following pseudo-code to describe the concrete steps taken to make one pilot’s schedule feasible.

Pseudo-code for the feasibility heuristic

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Racette, P., Quesnel, F., Lodi, A. et al. Gaining insight into crew rostering instances through ML-based sequential assignment. TOP (2024). https://doi.org/10.1007/s11750-024-00678-8

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DOI : https://doi.org/10.1007/s11750-024-00678-8

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Huo, L.; Xie, Y.; Li, J. An Innovative Deep Learning Futures Price Prediction Method with Fast and Strong Generalization and High-Accuracy Research. Appl. Sci. 2024 , 14 , 5602. https://doi.org/10.3390/app14135602

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