assignment method usage

The Assignment Method: Definition, Applications, and Implementation Strategies

Last updated 03/15/2024 by

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Understanding the assignment method

Optimized resource utilization, enhanced production efficiency, maximized profitability, applications of the assignment method, workforce allocation, production planning, sales territory management, resource budgeting.

• Optimizes resource utilization
• Enhances production efficiency
• Maximizes profitability
• Requires thorough analysis of past performance and market conditions
• Potential for misallocation of resources if not executed properly

Frequently asked questions

How does the assignment method differ from other resource allocation methods, what factors should organizations consider when implementing the assignment method, can the assignment method be applied to non-profit organizations or public sector agencies, what role does technology play in implementing the assignment method, are there any ethical considerations associated with the assignment method, key takeaways.

• The assignment method optimizes resource allocation to enhance efficiency and profitability.
• Applications include workforce allocation, production planning, sales territory management, and resource budgeting.
• Effective implementation requires thorough analysis of past performance and market conditions.
• Strategic allocation of resources can drive overall performance and revenue growth.

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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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Assignment Method

What is assignment method.

Assignment method is a way of allocating organizational resources where a resource is assigned to a particular task. The resource would be monetary, personnel, technological or another type of resource. The assignment method is used to determine what resources are assigned to which department, machine, or center of operation in the production process. This method is used to allocate the proper number of employees to a machine or task, and the number of jobs that a given machine or factory can produce. The idea is to assign resources in such a way that profits are maximized.

BREAKING DOWN Assignment Method

The assignment method is a way of allocating organizational resources to projects and tasks. The assignment method can be used for many other purposes besides production allocations. It can be employed to assign the number of salespersons to a given territory or territories. It can also be used to match bidders to contracts and assign other relevant components of business to each other. Regardless of the resource being allocated or the task to be accomplished, the idea is to assign resources in such a way that maximizes the amount of profit produced by the task or project.

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What is an Assignment Method?

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Assignment method.

In accounting and finance, the assignment method is a technique used for allocating or assigning resources, costs, or tasks among different departments, employees, or projects. The assignment method aims to optimize resource allocation to achieve maximum efficiency, cost savings, or other desired outcomes. It is often used in cost accounting, project management, and operations research.

For example, in cost accounting , the assignment method can be used to allocate indirect costs (such as overhead) to various cost centers or cost objects based on certain allocation criteria, like the proportion of direct labor hours or machine hours. This helps in determining the total cost of each product or service and aids in decision-making related to pricing, production levels, or resource allocation.

Another example is in project management, where the assignment method can be used to allocate tasks to team members based on their skills, availability, or other relevant factors. This helps in efficient task distribution, ensuring timely project completion, and optimal utilization of resources.

In summary, the assignment method is a technique used for allocating resources, costs, or tasks to optimize efficiency and achieve desired outcomes.

Example of an Assignment Method

Let’s take an example from cost accounting , specifically in a manufacturing company.

Suppose a manufacturing company produces three products: Product A, Product B, and Product C. The company has a total indirect overhead cost of $90,000. The indirect overhead cost needs to be allocated to each product based on machine hours used in production.

The machine hours used for each product are as follows:

• Product A: 600 hours
• Product B: 900 hours
• Product C: 1,500 hours

Total machine hours used: 3,000 hours (600 + 900 + 1,500)

Now, we will use the assignment method to allocate the indirect overhead costs based on the proportion of machine hours used for each product.

• Calculate the overhead rate per machine hour: \(\text{Overhead rate} = \frac{\text{Total overhead cost}}{\text{Total machine hours}} \) \(\text{Overhead rate} = \frac{90,000}{3,000 \text{ hours}} \) Overhead rate = $30 per machine hour
• Allocate the overhead cost to each product based on the machine hours used:
• Product A: 600 hours * $30 = $18,000
• Product B: 900 hours * $30 = $27,000
• Product C: 1,500 hours * $30 = $45,000

So, using the assignment method, the allocated overhead costs for Product A, Product B, and Product C are $18,000, $27,000, and $45,000, respectively. This allocation helps the company understand the total cost of producing each product and make informed decisions about pricing, production levels, and resource allocation.

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Chapter Questions

Use the assignment method to determine the best way to assign workers to jobs, given the following cost information. Compute the total cost for your assignment plan. $$ \begin{array}{|c|c|c|c|c|} \hline & & {3}{|c|}{\text { JOB }} \\ \hline & & \text { A } & \text { B } & \text { C } \\ \hline & 1 & 5 & 8 & 6 \\ \hline [t]{2}{*}{\text { Worker }} & 2 & 6 & 7 & g \\ \hline & 3 & 4 & 5 & 3 \\ \hline \end{array} $$

Rework Problem 1, treating the numbers in the table as profits instead of costs. Compute the total profit.

Assign trucks to delivery routes so that total costs are minimized, given the cost data shown. What is the total cost? $$ \begin{array}{ll|rrrrr} {7}{c}{\text { ROUTE }} \\ { 3 - 7 } & & {6}{c}{} \\ & & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ { 3 - 7 } & \mathbf{1} & 4 & 5 & 9 & 8 & 7 \\ & 2 & 6 & 4 & 8 & 3 & 5 \\ \text { Truck } & 3 & 7 & 3 & 10 & 4 & 6 \\ & 4 & 5 & 2 & 5 & 5 & 8 \\ & \mathbf{5} & 6 & 5 & 3 & 4 & 9 \end{array} $$

Develop an assignment plan that will minimize processing costs, given the information shown, and interpret your answer. $$ \begin{array}{cc|rrr} {1}{c}{} & {3}{c}{\text { WORKER }} \\ { 3 - 5 } & & \text { A } & \text { B } & \text { C } \\ { 3 - 5 } & & 8 & 11 \\ { 3 - 5 } \text { Job } & 1 & 12 & 10 & 8 \\ & 2 & 13 & 10 & 14 \\ & 3 & 14 & 9 & 12 \end{array} $$

Use the assignment method to obtain a plan that will minimize the processing costs in the following table under these conditions: a. The combination 2-D is undesirable. b. The combinations 1-A and 2-D are undesirable $$ \begin{aligned} &\text { WORKER }\\ &\begin{array}{cc|ccccc} { 3 - 6 } & & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} \\ { 3 - 6 } & \mathbf{1} & 14 & 18 & 20 & 17 & 18 \\ \text { Job } & \mathbf{2} & 14 & 15 & 19 & 16 & 17 \\ & 3 & 12 & 16 & 15 & 14 & 17 \\ & 4 & 11 & 13 & 14 & 12 & 14 \\ & \mathbf{5} & 10 & 16 & 15 & 14 & 13 \end{array} \end{aligned} $$

The following table contains information concerning four jobs that are awaiting processing at a work center. $$ \begin{array}{ccc} \text { Job } & \begin{array}{c} \text { Job Time } \\ \text { (days) } \end{array} & \begin{array}{c} \text { Due Date } \\ \text { (days) } \end{array} \\ \hline \text { A } & 14 & 20 \\ \text { B } & 10 & 16 \\ \text { C } & 7 & 15 \\ \text { D } & 6 & 17 \end{array} $$ a. Sequence the jobs using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. Assume the list is by order of arrival. b. For each of the methods in part $a$, determine (1) the average job flow time, (2) the average tardiness, and (3) the average number of jobs at the work center. c. Is one method superior to the others? Explain.

Using the information presented in the following table, identify the processing sequence that would result using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. For each method, determine (1) average job flow time, (2) average job tardiness, and (3) average number of jobs in the system. Jobs are listed in order of arrival. (Hint: First determine the total job time for each job by computing the total processing time for the job and then adding in the setup time. All times and due dates are in hours.)$$ \begin{array}{ccccr} \text { Job } & \begin{array}{c} \text { Processing } \\ \text { Time per Unit } \end{array} & \begin{array}{c} \text { Units } \\ \text { per Job } \end{array} & \begin{array}{c} \text { Setup } \\ \text { Time } \end{array} & \begin{array}{c} \text { Due } \\ \text { Date } \end{array} \\ \hline \text { a } & .14 & 45 & 0.7 & 4 \\ \text { b } & .25 & 14 & 0.5 & 10 \\ \text { c } & .10 & 18 & 0.2 & 12 \\ \text { d } & .25 & 40 & 1.0 & 20 \\ \text { e } & .10 & 75 & 0.5 & 15 \end{array} $$

The following table shows orders to be processed at a machine shop as of 8:00 a.m. Monday. The jobs have different operations they must go through. Processing times are in days. Jobs are listed in order of arrival. a. Determine the processing sequence at the first work center using each of these rules: (1) FCFS, (2) S/O. b. Compute the effectiveness of each rule using each of these measures: (1) average flow time, (2) average number of jobs at the work center. $$ \begin{array}{cccc} \text { Job } & \begin{array}{c} \text { Processing } \\ \text { Time } \\ \text { (days) } \end{array} & \begin{array}{c} \text { Due } \\ \text { Date } \\ \text { (days) } \end{array} & \begin{array}{c} \text { Remaining } \\ \text { Number of } \\ \text { Operations } \end{array} \\ \hline \text { A } & 8 & 20 & 2 \\ \text { B } & 10 & 18 & 4 \\ \text { C } & 5 & 25 & 5 \\ \text { D } & 11 & 17 & 3 \\ \text { E } & 9 & 35 & 4 \end{array} $$

A wholesale grocery distribution center uses a two-step process to fill orders. Tomorrow’s work will consist of filling the seven orders shown. Determine a job sequence that will minimize the time required to fill the orders. $$ \begin{aligned} &\text { TIME (hours) }\\ &\begin{array}{ccc} { 2 - 3 } \text { Order } & \text { Step 1 } & \text { Step 2 } \\ \hline \text { A } & 1.20 & 1.40 \\ \text { B } & 0.90 & 1.30 \\ \text { C } & 2.00 & 0.80 \\ \text { D } & 1.70 & 1.50 \\ \text { E } & 1.60 & 1.80 \\ \text { F } & 2.20 & 1.75 \\ \text { G } & 1.30 & 1.40 \end{array} \end{aligned} $$

The times required to complete each of eight jobs in a two-machine flow shop are shown in the table that follows. Each job must follow the same sequence, beginning with machine A and moving to machine B. a. Determine a sequence that will minimize makespan time. b. Construct a chart of the resulting sequence, and find machine B’s idle time. c. For the sequence determined in part a, how much would machine B’s idle time be reduced by splitting the last two jobs in half? $$ \begin{aligned} &\text { TIME (hours) }\\ &\begin{array}{ccc} { 2 - 3 } \text { Job } & \text { Machine A } & \text { Machine B } \\ \hline \text { a } & 16 & 5 \\ \text { b } & 3 & 13 \\ \text { c } & 9 & 6 \\ \text { d } & 8 & 7 \\ \text { e } & 2 & 14 \\ \text { f } & 12 & 4 \\ \text { g } & 18 & 14 \\ \text { h } & 20 & 11 \end{array} \end{aligned} $$

Given the operation times provided: a. Develop a job sequence that minimizes idle time at the two work centers. b. Construct a chart of the activities at the two centers, and determine each one’s idle time, assuming no other activities are involved. $$ \begin{aligned} &\text { JOB TIMES (minutes) }\\ &\begin{array}{lcccccc} & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \hline \text { Center 1 } & 20 & 16 & 43 & 60 & 35 & 42 \\ \text { Center 2 } & 27 & 30 & 51 & 12 & 28 & 24 \end{array} \end{aligned} $$

A shoe repair operation uses a two-step sequence that all jobs in a certain category follow. All jobs can be split in half at both stations. For the group of jobs listed, a. Find the sequence that will minimize total completion time. b. Determine the amount of idle time for workstation B. c. What jobs are candidates for splitting? Why? If they were split, how much would idle time and makespan time be reduced? $$ \begin{array}{lccccc} & {4}{c}{\text { JOB TIMES (minutes) }} \\ { 2 - 6 } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Workstation A } & 27 & 18 & 70 & 26 & 15 \\ \text { Workstation B } & 45 & 33 & 30 & 24 & 10 \end{array} $$

The following schedule was prepared by the production manager of Marymount Metal Shop: Determine a schedule that will result in earliest completion of all jobs on this list. $$ \begin{array}{cccccc} & {2}{c}{\text { CUTTING }} & & {2}{c}{\text { POLISHING }} \\ { 2 - 3 } { 5 - 6 } \text { Job } & \text { Start } & \text { Finish } & & \text { Start } & \text { Finish } \\ \hline \text { A } & 0 & 2 & & 2 & 5 \\ \text { B } & 2 & 6 & & 6 & 9 \\ \text { C } & 6 & 11 & & 11 & 13 \\ \text { D } & 11 & 15 & & 15 & 20 \\ \text { E } & 15 & 17 & & 20 & 23 \\ \text { F } & 17 & 20 & & 23 & 24 \\ \text { G } & 20 & 21 & & 24 & 28 \end{array} $$

The production manager must determine the processing sequence for seven jobs through the grinding and deburring departments. The same sequence will be followed in both departments. The manager’s goal is to move the jobs through the two departments as quickly as possible. The foreman of the deburring department wants the SPT rule to be used to minimize the work-in- process inventory in his department. $$ \begin{array}{ccc} &{2}{c}{\begin{array}{c} \text { PROCESSING TIME } \\ \text { (hours) } \end{array}} \\ { 2 - 3 } \text { Job } & \text { Grinding } & \text { Deburring } \\ \hline \text { A } & 3 & 6 \\ \text { B } & 2 & 4 \\ \text { C } & 1 & 5 \\ \text { D } & 4 & 3 \\ \text { E } & 9 & 4 \\ \text { F } & 8 & 7 \\ \text { G } & 6 & 2 \end{array} $$ a. Prepare a schedule using SPT for the grinding department. b. What is the flow time in the grinding department for the SPT sequence? What is the total time needed to process the seven jobs in both the grinding and deburring departments? c. Determine a sequence that will minimize the total time needed to process the jobs in both departments. What flow time will result for the grinding department? d. Discuss the trade-offs between the two alternative sequencing arrangements. At what point would the production manager be indifferent concerning the choice of sequences?

A foreman has determined processing times at a work center for a set of jobs and now wants to sequence them. Given the information shown, do the following: a. Determine the processing sequence using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. For each sequence, compute the average job tardiness, the average flow time, and the average number of jobs at the work center. The list is in FCFS order. b. Using the results of your calculations in part a, show that the ratio of average flow time and the average number of jobs measures are equivalent for all four sequencing rules. c. Determine the processing sequence that would result using the S/O rule. $$ \begin{array}{cccc} \text { Job } & \begin{array}{c} \text { Job Time } \\ \text { (days) } \end{array} & \begin{array}{c} \text { Due } \\ \text { Date } \end{array} & \begin{array}{c} \text { Operations } \\ \text { Remaining } \end{array} \\ \hline \text { a } & 4.5 & 10 & 3 \\ \text { b } & 6.0 & 17 & 4 \\ \text { c } & 5.2 & 12 & 3 \\ \text { d } & 1.6 & 27 & 5 \\ \text { e } & 2.8 & 18 & 3 \\ \text { f } & 3.3 & 19 & 1 \end{array} $$

Given the information in the following table, determine the processing sequence that would result using the S/O rule. $$ \begin{array}{cccc} \text { Job } & \begin{array}{c} \text { Remaining } \\ \text { Processing } \\ \text { Time (days) } \end{array} & \begin{array}{c} \text { Due } \\ \text { Date } \end{array} & \begin{array}{c} \text { Remaining } \\ \text { Number of } \\ \text { 0perations } \end{array} \\ \hline \text { a } & 5 & 8 & 2 \\ \text { b } & 6 & 5 & 4 \\ \text { c } & 9 & 10 & 4 \\ \text { d } & 7 & 12 & 3 \\ \text { e } & 8 & 10 & 2 \end{array} $$

Given the following information on job times and due dates, determine the optimal processing sequence using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. For each method, find the average job flow time and the average job tardiness. Jobs are listed in order of arrival. $$ \begin{array}{ccc} \text { Job } & \begin{array}{c} \text { Job Time } \\ \text { (hours) } \end{array} & \begin{array}{c} \text { Due Date } \\ \text { (hours) } \end{array} \\ \hline \text { a } & 3.5 & 7 \\ \text { b } & 2.0 & 6 \\ \text { c } & 4.5 & 18 \\ \text { d } & 5.0 & 22 \\ \text { e } & 2.5 & 4 \\ \text { f } & 6.0 & 20 \end{array} $$

The Budd Gear Co. specializes in heat-treating gears for automobile companies. At 8:00 a.m., when Budd's shop opened today, five orders (listed in order of arrival) were waiting to be processed. $$ \begin{array}{cccc} \text { Order } & \begin{array}{c} \text { Order Size } \\ \text { (units) } \end{array} & \begin{array}{c} \text { Per Unit Time in } \\ \text { Heat Treatment } \\ \text { (minutes/unit) } \end{array} & \begin{array}{c} \text { Due Date } \\ \text { (min. from now) } \end{array} \\ \hline \text { A } & 16 & 4 & 160 \\ \text { B } & 6 & 12 & 200 \\ \text { C } & 10 & 3 & 180 \\ \text { D } & 8 & 10 & 190 \\ \text { E } & 4 & 1 & 220 \end{array} $$

The following table contains order-dependent setup times for three jobs. Which processing sequence will minimize the total setup time? $$ \begin{array}{lccccc} & & &{2}{c}{\begin{array}{c} \text { Following Job's } \\ \text { Setup Time (hrs.) } \end{array}} \\ & & \begin{array}{c} \text { Setup } \\ \text { Time (hrs.) } \end{array} & \text { A } & \text { B } & \text { C } \\ \hline \text { Preceding } & \text { A } & 2 & - & 3 & 5 \\ \text { Job } & \text { B } & 3 & 8 & - & 2 \\ & \text { C } & 2 & 4 & 3 & - \end{array} $$

The following table contains order-dependent setup times for three jobs. Which processing sequence will minimize the total setup time? $$ \begin{array}{|c|c|c|c|c|c|} \hline & & [b]{2}{*}{\begin{array}{c} \text { Setup } \\ \text { Time (hrs.) } \end{array}} & {3}{|c|}{\begin{array}{l} \text { Following Job's Setup } \\ \text { Time (hrs.) } \end{array}} \\ \hline {4}{*}{\begin{array}{l} \text { Preceding } \\ \text { Job } \end{array}} & & & \text { A } & \text { B } & \text { C } \\ \hline & \text { A } & 2.4 & - & 1.8 & 2.2 \\ \hline & \text { B } & 3.2 & 0.8 & - & \\ \hline & \text { C } & 2.0 & 2.6 & 1.3 & \\ \hline \end{array} $$

The following table contains order-dependent setup times for four jobs. For safety reasons, job C cannot follow job A, nor can job A follow job C. Determine the processing sequence that will minimize the total setup time. ( Hint: There are 12 alternatives.) $$ \begin{array}{|c|c|c|c|c|c|c|} \hline & & [b]{2}{*}{\begin{array}{c} \text { Setup } \\ \text { Time (hrs.) } \end{array}} & {4}{|c|}{\begin{array}{l} \text { Following Job's } \\ \text { Setup Time (hrs.) } \end{array}} \\ \hline & & & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline & \text { A } & 2 & - & 5 & x & 4 \\ \hline \text { Preceding } & \text { B } & 1 & 7 & - & 3 & 2 \\ \hline [t]{2}{*}{\text { Job }} & \text { C } & 3 & x & 2 & \text { - } & 2 \\ \hline & \text { D } & 2 & 4 & 3 & 6 & - \\ \hline \end{array} $$

Given this information on planned and actual inputs and outputs for a service center, determine the work backlog for each period. The beginning backlog is 12 hours of work. The figures shown are standard hours of work. $$ \begin{aligned} &\text { PERIOD }\\ &\begin{array}{|l|l|c|c|c|c|c|} \hline {4}{*}{\text { Input }} & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ { 2 - 7 } & \text { Planned } & 24 & 24 & 24 & 24 & 20 \\ \hline \text { Actual } & 25 & 27 & 20 & 22 & 24 \\ \hline \end{array}\\ &\begin{array}{|c|c|c|c|c|c|c|} \hline [t]{2}{*}{\text { Output }} & \text { Planned } & 24 & 24 & 24 & 24 & 23 \\ \hline & \text { Actual } & 24 & 22 & 23 & 24 & 24 \\ \hline \end{array} \end{aligned} $$

Given the following data on inputs and outputs at a work center, determine the cumulative deviation and the backlog for each time period. The beginning backlog is 7. $$ \begin{aligned} &\text { PERIOD }\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline \text { Planned } & 200 & 200 & 180 & 190 & 190 & 200 \\ \hline \text { Actual } & 210 & 200 & 179 & 195 & 193 & 194 \\ \hline \end{array} \end{aligned} $$ $$ \begin{array}{|l|l|c|c|c|c|c|c|} \hline & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline {2}{*}{\text { Output }} & \text { Planned } & 200 & 200 & 180 & 190 & 190 & 200 \\ { 2 - 7 } & \text { Actual } & 205 & 194 & 177 & 195 & 193 & 200 \\ { 2 - 7 } & & & & & & & \end{array} $$

Determine the minimum number of workers needed, and a schedule for the following staffing requirements, giving workers two consecutive days off per cycle (not including Sunday). $$ \begin{array}{lcccccc} \text { Day } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thu } & \text { Fri } & \text { Sat } \\ \hline \text { Staff needed } & 2 & 3 & 1 & 2 & 4 & 3 \end{array} $$

Determine the minimum number of workers needed, and a schedule for the following staffing requirements, giving workers two consecutive days off per cycle (not including Sunday). $$ \begin{array}{lcccccr} \text { Day } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thu } & \text { Fri } & \text { Sat } \\ \hline \text { Staff needed } & 3 & 4 & 2 & 3 & 4 & 5 \end{array} $$

Determine the minimum number of workers needed, and a schedule for the following staffing requirements, giving workers two consecutive days off per cycle (not including Sunday)$$ \begin{array}{lcccccc} \text { Day } & \text { Mon } & \text { Tue } & \text { Wed } & \text { Thu } & \text { Fri } & \text { Sat } \\ \hline \text { Staff needed } & 4 & 4 & 5 & 6 & 7 & 8 \end{array} $$

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:

Variations of the Assignment Problems:

Unbalanced Assignment Problem:

Any assignment problem is said to be unbalanced if the cost matrix is not a square matrix, i.e. the no of rows and the no of columns are not equal. To make it balanced we add a dummy row or dummy column with all the entries is zero.

There are four jobs to be assigned to the machines. Only one job could be assigned to one machine are given in following matrix.

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

Python Tutorial

File handling, python modules, python numpy, python pandas, python matplotlib, python scipy, machine learning, python mysql, python mongodb, python reference, module reference, python how to, python examples, python assignment operators.

Assignment operators are used to assign values to variables:

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Variable assignment within method parameter

I've just found out (by discovering a bug) that you can do this:

Is there a legitimate use of using the returned value of an assignment?

(Obviously i++ is, but is this the same?)

• 3 I'm not quite sure how this is a bug. –  Arran Commented Sep 24, 2013 at 10:15
• 3 See Eric Lippert's answer here stackoverflow.com/a/3807583/187697 –  keyboardP Commented Sep 24, 2013 at 10:25
• 2 @Arran The bug was that somebody had copied and pasted the signature of a method with an optional parameter and forgot to get rid of the default, causing undesired behaviour. No bugs in the code quoted here. –  dav_i Commented Sep 24, 2013 at 10:26

5 Answers 5

Generally I'd avoid using the return value of an assignment as it can all too easily lead to had to spot bugs. However, there is one excellent use for the feature as hopefully illustrated below, lazy initialization:

As of C# 6, this can be expressed using the => notation:

By using the ?? notation and the return value from the assignment, terse, yet readable, syntax can be used to only initialize the field and return it via a property when that property is called. In the above example, this isn't so useful, but within eg facades that need to be unit tested, only initializing those parts under test can greatly simplify the code.

• 1 Interesting example. Are you missing return though? –  dav_i Commented Sep 24, 2013 at 10:29
• 1 +1 for the great idea. need a pair of parenthesis to compile. public string Value { get { return _value ?? (_value = "hello"); } } –  Menol Commented Oct 14, 2016 at 8:32
• 1 @Menol, thanks for spotting that.I've also updated it to mention the optional syntax that can be used with C# 6. –  David Arno Commented Oct 14, 2016 at 8:42

This is legitimate.

s = "hello" , is an expression which is evaluated / executed first, and the int.TryParse expression is executed after that.

Therefore, int.TryParse will use the content of 's' which is at that time "hello" and it's returning false.

An assignment is an expression just like any other. This is valid syntax.

For the same reason this is valid:

From = Operator (C# Reference)

The assignment operator (=) stores the value of its right-hand operand in the storage location, property, or indexer denoted by its left-hand operand and returns the value as its result.

That means = doesn't do assignment only, also it returns the value as an expression . Inside a method, your s reference now points to "hello" string not "3" anymore..

is evaluated like;

which since "hello" is not a valid integer, it returns false .

You are basically asking:

Is it useful that assignment to a variable can be used as a value?

One place where it is useful is for daisy-chaining variable assignments:

Which is clearer when parenthesized:

If c = "hello" did not have the value hello , then the above would not be possible. It has limited use elsewhere but has no real downside.

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MsoAssignmentMethod enumeration (Office)

• 5 contributors

Indicates the assignment method in a LabelInfo object.

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Distributed Combinatorial Optimization of Downlink User Assignment in mmWave Cell-free Massive MIMO Using Graph Neural Networks

• Besser, Karl-Ludwig
• Raghunath, Ramprasad
• Schmeink, Anke
• Jorswieck, Eduard A
• Caire, Giuseppe
• Poor, H. Vincent

Millimeter wave (mmWave) cell-free massive MIMO (CF mMIMO) is a promising solution for future wireless communications. However, its optimization is non-trivial due to the challenging channel characteristics. We show that mmWave CF mMIMO optimization is largely an assignment problem between access points (APs) and users due to the high path loss of mmWave channels, the limited output power of the amplifier, and the almost orthogonal channels between users given a large number of AP antennas. The combinatorial nature of the assignment problem, the requirement for scalability, and the distributed implementation of CF mMIMO make this problem difficult. In this work, we propose an unsupervised machine learning (ML) enabled solution. In particular, a graph neural network (GNN) customized for scalability and distributed implementation is introduced. Moreover, the customized GNN architecture is hierarchically permutation-equivariant (HPE), i.e., if the APs or users of an AP are permuted, the output assignment is automatically permuted in the same way. To address the combinatorial problem, we relax it to a continuous problem, and introduce an information entropy-inspired penalty term. The training objective is then formulated using the augmented Lagrangian method (ALM). The test results show that the realized sum-rate outperforms that of the generalized serial dictatorship (GSD) algorithm and is very close to the upper bound in a small network scenario, while the upper bound is impossible to obtain in a large network scenario.

• Electrical Engineering and Systems Science - Signal Processing