An Assignment Problem and Its Application in Education Domain: A Review and Potential Path
- Advances in Operations Research 2018(45-48):1-19
- 2018(45-48):1-19
- Universiti Utara Malaysia
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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
- 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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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:
Example: In the NRMP, medical residents submit a
Residency Program Assignments - Assignment problem applications: Real world scenarios and solutions
Sports scheduling is a crucial aspect of organizing tournaments and leagues. It is not just about arranging games, but rather about creating fair and balanced matchups. The biggest challenge is to maintain an equitable balance between the teams while ensuring that each team plays against all the others within a given time frame. It requires thoughtful planning, strategic scheduling, and the use of advanced algorithms. In this section, we will explore the complexities of sports scheduling and how different approaches can be used to create fair and balanced matchups.
1. Round Robin Scheduling: This is a common scheduling method that ensures each team plays against all others in a league. It is a straightforward approach where each team plays a fixed number of games against the others. It is simple, but it doesn't guarantee fairness. For example, a team may have to play against a stronger opponent more times than a weaker one, giving them a disadvantage.
2. Power Ranking: Power ranking is a method of ranking teams based on their performance in previous games. It is a way to create a hierarchy of teams and use it to schedule games. This approach pairs strong teams against one another and weak teams against each other, ensuring balance in each matchup. However, it can be challenging to determine an accurate power ranking, especially in the early stages of a tournament.
3. Double Round Robin: This is another scheduling method where each team plays against every other team twice, once at home and once away. It is a more comprehensive approach than round-robin scheduling, but it requires more time and resources. It is an effective way to ensure balance in matchups, as each team plays against every other team an equal number of times.
4. Travel Considerations: For tournaments that involve teams from different regions or countries, travel considerations must be taken into account. It is essential to minimize travel time and expenses while ensuring that each team gets a fair share of home and away games. One way to do this is to schedule games in clusters, where teams play against multiple opponents in one city before moving on to the next.
Sports scheduling is a complex task that requires careful planning and execution. Creating fair and balanced matchups is crucial, and it can be achieved through the use of different scheduling methods such as round-robin, power ranking, and double round-robin. By taking into account travel considerations and using advanced algorithms, it is possible to create a tournament schedule that is both fair and efficient.
Creating Fair and Balanced Matchups - Assignment problem applications: Real world scenarios and solutions
supply chain optimization is a critical aspect of any business that involves the movement of goods from suppliers to customers. It aims to streamline the entire supply chain process, from inventory management to distribution center allocation, in order to maximize efficiency and minimize costs . Allocating inventory and distribution centers effectively is a key component of supply chain optimization, as it directly impacts the speed and accuracy of order fulfillment .
From the perspective of suppliers, efficient inventory allocation ensures that they have the right amount of stock available at the right time. This prevents overstocking or stockouts, both of which can lead to financial losses. By accurately forecasting demand and strategically placing inventory in different locations, suppliers can optimize their operations and meet customer demands more effectively.
On the other hand, from the viewpoint of customers, efficient distribution center allocation plays a crucial role in ensuring timely delivery and reducing shipping costs. By strategically locating distribution centers closer to major markets or areas with high customer demand, businesses can minimize transportation time and costs. This not only improves customer satisfaction but also enhances overall supply chain performance.
To achieve optimal inventory and distribution center allocation, businesses can leverage various techniques and tools. Here are some key strategies:
1. demand forecasting : Accurate demand forecasting is essential for determining the appropriate level of inventory required at different locations. By analyzing historical data , market trends, and customer behavior patterns , businesses can make informed decisions about how much stock should be allocated to each distribution center.
For example, an e-commerce company may use predictive analytics to forecast demand during peak shopping seasons like Black Friday or Christmas. Based on these forecasts, they can allocate additional inventory to distribution centers located near regions with high expected demand.
2. network optimization : Network optimization involves evaluating different scenarios and configurations for distribution centers based on factors such as transportation costs, lead times, and service levels. By modeling different network designs using mathematical algorithms or simulation software, businesses can identify the most cost-effective allocation strategy.
For instance, a multinational retail chain may use network optimization to determine the optimal number and location of distribution centers across different regions. This analysis considers factors like transportation costs, customer density, and proximity to suppliers to identify the most efficient allocation strategy.
3. inventory management systems : Implementing advanced inventory management systems can greatly enhance supply chain optimization efforts. These systems use real-time data and algorithms to monitor inventory levels, track demand patterns, and automatically trigger replenishment orders when necessary.
For example, a manufacturer may utilize an inventory management system that integrates with their suppliers' systems. This allows them to automatically
Allocating Inventory and Distribution Centers - Assignment problem applications: Real world scenarios and solutions
School bus routing plays a crucial role in efficiently assigning routes for student transportation. It involves the complex task of determining the most optimal routes for buses to pick up and drop off students, taking into consideration factors such as distance, time, safety, and cost. This section will delve into the various aspects of school bus routing, exploring its challenges, solutions, and real-world applications .
1. Importance of Efficient School Bus Routing:
Efficient school bus routing is essential for ensuring that students are transported to and from school in a timely manner while minimizing costs and maximizing safety. By optimizing routes, schools can reduce travel time, fuel consumption, and vehicle wear and tear. Moreover, efficient routing helps to minimize the number of buses required, leading to cost savings for schools or districts.
2. Challenges in School Bus Routing:
Routing school buses presents several challenges due to the complexity of the problem. Factors such as varying student populations, changing addresses, traffic conditions, road networks, and diverse scheduling requirements make it difficult to create an optimal routing plan manually. Additionally, constraints like capacity limitations of buses and adherence to state regulations further complicate the process.
3. Solutions for Efficient School Bus Routing:
To overcome the challenges associated with school bus routing, advanced algorithms and technologies have been developed. These solutions leverage mathematical optimization techniques to automate the process and generate optimal routes based on predefined criteria . For instance, algorithms like the traveling Salesman problem (TSP) or Vehicle Routing Problem (VRP) can be applied to find the shortest or most efficient routes for buses.
4. Real-World Applications:
Efficient school bus routing has been successfully implemented in various real-world scenarios. For example, a school district in California utilized a routing software that considered factors like traffic patterns and student locations to optimize their bus routes. As a result, they reduced their fleet size by 20% while maintaining punctuality and saving significant costs on fuel.
5. Benefits of Efficient School Bus Routing:
Implementing efficient school bus routing brings numerous benefits. It not only reduces transportation costs but also minimizes the environmental impact by optimizing fuel consumption and vehicle emissions. Moreover, it enhances student safety by ensuring that buses follow the most secure routes and adhere to traffic regulations.
6. Future Trends in School Bus Routing:
As technology continues to advance, the future of school bus routing holds promising developments. Integration with GPS tracking systems, real-time data analysis , and machine learning algorithms can further enhance efficiency and adaptability in routing plans. These advancements will enable schools to respond
Efficiently Assigning Routes for Student Transportation - Assignment problem applications: Real world scenarios and solutions
emergency response planning is a critical aspect of any organization's preparedness for crisis situations. When faced with an emergency, it is essential to have a well-defined plan in place that outlines how resources will be assigned and utilized effectively. This section delves into the intricacies of assigning resources during crisis situations, exploring different perspectives and providing in-depth insights on this crucial aspect of emergency response planning.
1. Understanding the Importance of Resource Assignment:
Assigning resources during crisis situations is vital as it ensures that the right people, equipment, and materials are deployed to address the emergency effectively. By having a clear understanding of available resources and their capabilities, organizations can optimize their response efforts and minimize potential risks.
For example, in the case of a natural disaster such as a hurricane, assigning resources may involve deploying search and rescue teams, medical personnel, and supplies to affected areas. By carefully assessing the situation and allocating resources accordingly, emergency responders can provide timely assistance to those in need.
2. Assessing Resource Availability:
Before assigning resources, it is crucial to assess their availability accurately. This includes evaluating the number of personnel, equipment, and supplies that can be mobilized within a given timeframe. Organizations must consider factors such as geographical location, operational capacity, and logistical constraints when determining resource availability.
For instance, during a large-scale fire incident, fire departments need to evaluate the number of available firefighters, fire engines, and water supply sources. By understanding these limitations upfront, they can make informed decisions about resource allocation and request additional support if necessary.
3. Prioritizing Resource Allocation:
In crisis situations where resources may be limited or overwhelmed by demand, prioritization becomes paramount. Assigning resources based on urgency and criticality ensures that immediate needs are addressed first while still considering long-term requirements.
Consider a mass casualty incident where multiple injured individuals require medical attention simultaneously. In such cases, triage systems are employed to prioritize patients based on the severity of their injuries. This allows medical personnel to allocate resources efficiently , ensuring that those in critical condition receive immediate care.
4. Coordinating Resource Assignment:
Effective coordination is essential when assigning resources during crisis situations. It involves establishing clear communication channels , defining roles and responsibilities , and fostering collaboration among different response teams and agencies.
For example, in the event of a terrorist attack, law enforcement agencies , emergency medical services , and fire departments must work together seamlessly. By coordinating their efforts and sharing information, they can allocate resources strategically to neutralize threats, provide medical assistance, and ensure public safety
Assigning Resources during Crisis Situations - Assignment problem applications: Real world scenarios and solutions
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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey
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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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- DOI: 10.1155/2018/8958393
- Corpus ID: 49679551
An Assignment Problem and Its Application in Education Domain: A Review and Potential Path
- Syakinah Faudzi , Syariza Abdul-Rahman , R. A. Rahman
- Published in Advances in Operations… 17 May 2018
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A Comparative Analysis of Assignment Problem
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- Shahriar Tanvir Alam ORCID: orcid.org/0000-0002-0567-3381 5 ,
- Eshfar Sagor 5 ,
- Tanjeel Ahmed 5 ,
- Tabassum Haque 5 ,
- Md Shoaib Mahmud 5 ,
- Salman Ibrahim 5 ,
- Ononya Shahjahan 5 &
- Mubtasim Rubaet 5
Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))
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The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.
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Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11
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The formal definition of the assignment problem (or linear assignment problem) is . Given two sets, A and T, together with a weight function C : A × T → R.Find a bijection f : A → T such that the cost function: (, ())is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as: , The problem is "linear" because the cost ...
302 Applications of Discrete Mathematics permutation can be generated in just 10−9 seconds, an assignment problem with n = 30 would require at least 8· 1015 years of computer time to solve by generating all 30! permutations. Therefore a better method is needed. Before developing a better algorithm, we need to set up a model for the
This paper presents a review pertaining to assignment problem within the education domain, besides looking into the applications of the present research trend, developments, and publications ...
Step 3: Cover all zeroes with minimum number of. horizontal and vertical lines. Step 4: Since we need 3 lines to cover all zeroes, we have found the optimal assignment. 2500 4000 3500. 4000 6000 3500. 2000 4000 2500. So the optimal cost is 4000 + 3500 + 2000 = 9500. An example that doesn't lead to optimal value in first attempt: In the above ...
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 ...
Special Types of Linear Programming Problems. Bernard Kolman, Robert E. Beck, in Elementary Linear Programming with Applications (Second Edition), 1995. 5.2 THE ASSIGNMENT PROBLEM. We gave an example of the assignment problem in Section 4.1., Example 3.In this section we will discuss the simplest formulation of the problem.
The problem of optimally assigning m individuals to m jobs, so that each individual is assigned to one job, and each job is filled by one individual. The problem can be formulated as a linear-programming problem with the objective function measuring the (linear) utility of the assignment as follows:
Abstract. Having reached the 50th (golden) anniversary of the publication of Kuhn's seminal article on the solution of the classic assignment problem, it seems useful to take a look at the variety of models to which it has given birth. This paper is a limited survey of what appear to be the most useful of the variations of the assignment ...
Assignment problems involve optimally matching the elements of two or more sets, where the dimension of the problem refers to the number of sets of elements to be matched. ... Modern IoT applications, such as smart surveillance systems, remote health monitoring applications, ambient-aware applications etc., are time-critical and computationally ...
Assignment problems deal with the question of how to assign other items (machines, tasks). There are different ways in mathematics to describe an assignment: we can view an assignment as a bijective mapping φ between two finite sets . We can write a permutation φ as 12…nφ (1)φ (2)…φ (n), which means that 1 is mapped to φ (1), 2 is ...
The quadratic assignment problem (QAP) has considered one of the most significant combinatorial optimization problems due to its variant and significant applications in real life such as scheduling, production, computer manufacture, chemistry, facility location, communication, and other fields. QAP is NP-hard problem that is impossible to be solved in polynomial time when the problem size ...
The assignment problem is a classic optimization problem that has found numerous applications in various fields. From logistics and transportation to scheduling and resource allocation, the assignment problem offers practical solutions to real-world scenarios where tasks need to be assigned to resources or individuals in the most efficient ...
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 ...
Natural applications.! Match jobs to machines.! Match personnel to tasks.! Match PU students to writing seminars. Non-obvious applications.! Vehicle routing.! ... Equivalent Assignment Problem c(x, y) 00312 01015 43330 00110 12204 cp(x, y) 3891510 41071614 913111910 813122013 175119 8 13 11 19 13 5 4 3 0 8 9 + 8 - 13 10
5. Identify several areas of application of transportation problems and their variants. 6. Describe the characteristics of assignment problems. 7. Identify the relationship between assignment problems and transportation problems. 8. Formulate a spreadsheet model for an assignment problem from a description of the problem. 9.
•Examples of assignment problems •Assignment Algorithms Auction and variants Hungarian Algorithm (also called Kuhn-Munkres Algorithm) Easy to understand, but not for practical applications Successive shortest path algorithm (Hungarian; Jonker, Volgenant and Castanon (JVC)) Signature … Not efficient computationally •Special cases •M ...
The Assignment Problem: An Example A company has 4 machines available for assignment to 4 tasks. Any machine can be assigned to any task, and each task requires processing by one machine. The time required to set up each machine for the processing of each task is given in the table below. TIME (Hours) Task 1 Task 2 Task 3 Task 4 Machine 1 13 4 7 6
This review summarizes and records a comprehensive survey regarding assignment problem within education domain, which enhances one's understanding concerning the varied types of assignment problems, along with various approaches that serve as solution. This paper presents a review pertaining to assignment problem within the education domain, besides looking into the applications of the ...
2.1.1 The Linear Assignment Problem A commonly used intuitive introduction to the assignment problem as used by Hanan and Kurtzberg [25], involves the assignment or n people to n jobs. For each job assignment, there is a related cost, c ij, of assigning person i to job j. The objective is to assign each person to one and only one job in such
Section 6.5 : Applications. We have $2,500 to invest and 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded. We are starting with $60,000 and we're going to put it into an account that earns an annual interest rate of 7.5%.
Assignment Problems Rainer Burkard Graz University of Technology Graz, Austria Mauro Dell'Amico University of Modena and Reggio Emilia ... 3.8 Applications of maximum matching problems ..... 65 3.8.1 Vehicle scheduling problems ..... 65 3.8.2 Time slot assignment problem ..... 66 vii. book 2012/2/21 page viii viii Contents ...
Assignment problems deal with the question how to assign n items (e.g. jobs) to n machines (or workers) in the best possible way. ... An algorithm for solving 3-dimensional assignment problems with application to scheduling in a teaching practice, Journal of the Operational Research Society 32, 1981, 989-995. ...
Tables 2, 3, 4, and 5 present the steps required to determine the appropriate job assignment to the machine. Step 1 By taking the minimum element and subtracting it from all the other elements in each row, the new table will be: Table 2 represents the matrix after completing the 1st step. Table 1 Initial table of a.