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:
Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.
The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:
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- 2. It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while minimizing the total assignment costs. One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project). An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.
- 3. This method was developed by D. Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem. In order to use this method, one needs to know only the cost of making all the possible assignments. Each assignment problem has a matrix (table) associated with it. Normally, the objects (or people) one wishes to assign are expressed in rows, whereas the columns represent the tasks (or things) assigned to them. The number in the table would then be the costs associated with each particular assignment.
- 4. Though assignment problem finds applicability in various diverse business situations, we discuss some of its main application areas: In assigning machines to factory orders. In assigning sales/marketing people to sales territories. In assigning contracts to bidders by systematic bid-evaluation. In assigning teachers to classes. In assigning accountants to accounts of the clients.
- 5. Mathematically the assignment problem can be expressed as The objective function is Minimize C11X11 + C12X12 + ------- + CnnXnn.
- 6. Step 1. Determine the cost table from the given problem. (i) If the no. of sources is equal to no. of destinations, go to step 3. (ii) If the no. of sources is not equal to the no. of destination, go to step2. Step 2. Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero. Step 3. Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row.
- 7. Step 4. In the reduced matrix obtained in the step 3, locate the smallest element of each column and then subtract the same from each element of that column. Each column and row now have at least one zero. Step 5. In the modified matrix obtained in the step 4, search for the optimal assignment as follows: (a) Examine the rows successively until a row with a single zero is found. Enrectangle this row ( )and cross off (X) all other zeros in its column. Continue in this manner until all the rows have been taken care of. (b) Repeat the procedure for each column of the reduced matrix. (c) If a row and/or column has two or more zeros and one cannot be chosen by inspection then assign arbitrary any one of these zeros and cross off all other zeros of that row / column. (d) Repeat (a) through (c) above successively until the chain of assigning ( ) or cross (X) ends.
- 8. Step 6. If the number of assignment ( ) is equal to n (the order of the cost matrix), an optimum solution is reached. If the number of assignment is less than n(the order of the matrix), go to the next step. Step7. Draw the minimum number of horizontal and/or vertical lines to cover all the zeros of the reduced matrix. Step 8. Develop the new revised cost matrix as follows: (a)Find the smallest element of the reduced matrix not covered by any of the lines. (b)Subtract this element from all uncovered elements and add the same to all the elements laying at the intersection of any two lines. Step 9. Go to step 6 and repeat the procedure until an optimum solution is attained.
- 9. A job has four men available for work on four separate jobs. Only one man can work on any one job. The cost of assigning each man to each job is given in the following table. The objective is to assign men to jobs such that the total cost of assignment is minimum.
- 10. Step 1 Identify the minimum element in each row and subtract it from every element of that row.
- 11. Step 2 Identify the minimum element in each column and subtract it from every element of that column.
- 12. Make the assignment for the reduced matrix obtain from steps 1 and 2 in the following way:
- 13. Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix obtained from last step
- 14. Select the smallest element from all the uncovered elements. Subtract this smallest element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment.
- 15. Since the number of assignments is equal to the number of rows (& columns), this is the optimal solution. The total cost of assignment = A1 + B4 + C2 + D3 Substitute the values from original table: 20 + 17 + 24 + 17 = 78.
- 16. Some assignment problems entail maximizing the profit, effectiveness, or layoff of an assignment of persons to tasks or of jobs to machines. The conversion is accomplished by subtracting all the elements of the given effectiveness matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.
- 17. Five different machines can do any of the five required jobs, with different profits resulting from each assignment as given below:
- 18. Here, the highest element is 62. So we subtract each value from 62. The maximum profit through this assignment is 214.
- 19. It is an assignment problem where the number of persons is not equal to the number of jobs. If the number of persons is less than the number of jobs then we introduce one or more dummy persons (rows) with zero values to make the assignment problem balanced. Likewise, if the number of jobs is less than the number of persons then we introduce one or more dummy jobs (columns) with zero values to make the assignment problem balanced
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Six Characteristics of a Model Assignment
How many times have you had a student submit an assignment with few sources, poorly written and several days late? Probably happens more times than not. There are six characteristics of a model assignment which will not only alleviate instructor frustration, but also strengthen student writing and time management skills.
- Create assignments which directly relate to accomplishing the course objective. A model assignment maintains a clear goal toward accomplishing a course objective. For adult online learners, course goals relate less to theory or original research and more to practical approaches for day-to-day application or career advancement.
- More details equals higher quality of student final product. Since adult online learners come from diverse backgrounds, do not assume students will understand the purpose of the assignment. Be prepared to tell students what you expect (e.g. word count, citation format, number of sources, etc.) and how it should be done (e.g. upload to Moodle versus email attachment).
- Give incremental due dates. Large comprehensive assignments due at the course finality leads to unfocused, or even plagiarized, writing. Break down a large assignment into several smaller assignments due sporadically throughout the term. In turn, students receive valuable feedback incrementally as they progress throughout the course.
- Allow students to brainstorm for topics. Allow students to brainstorm topics or share with other students using the Moodle Discussion Board form. Or consider offering students a choice among 3-4 essay questions, case scenarios, or case studies. By allowing student choice, students will find a greater connection in their writing which in turn will lead to better final submissions.
- Give examples. In addition to clear directions, students also appreciate a visual piece of the final product. If you decide to use another student’s work, be sure to ask permission to use from the student. Post model assignments on your Moodle course shell.
- Share student evaluation tools. Share rubrics, or other evaluation tool, early in the assignment rather than at the end so students may clarify expectations firsthand. Post rubrics or evaluation tools on your Moodle course shell so students may refer to it when necessary.
Assignment Problem: Linear Programming
The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.
In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.
The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.
It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.
"A mathematician is a device for turning coffee into theorems." -Paul Erdos
Formulation of an assignment problem
Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:
The structure of an assignment problem is identical to that of a transportation problem.
To formulate the assignment problem in mathematical programming terms , we define the activity variables as
for i = 1, 2, ..., n and j = 1, 2, ..., n
In the above table, c ij is the cost of performing jth job by ith worker.
Generalized Form of an Assignment Problem
The optimization model is
Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn
subject to x i1 + x i2 +..........+ x in = 1 i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1 j = 1, 2,......., n
x ij = 0 or 1
In Σ Sigma notation
x ij = 0 or 1 for all i and j
An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.
Assumptions in Assignment Problem
- Number of jobs is equal to the number of machines or persons.
- Each man or machine is assigned only one job.
- Each man or machine is independently capable of handling any job to be done.
- Assigning criteria is clearly specified (minimizing cost or maximizing profit).
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Operations Research Simplified Back Next
Goal programming Linear programming Simplex Method Transportation Problem
What is an Assignment Model? (Sirug, 2012) · It is a special case of transportation problem that requires pairing of various items to various receivers in such a way that the total cost/profit of the pairings is minimized or maximized. Difference between Transportation and Assignment Model
Application Area of Assignment Model
· Assigning teachers to classes · Assigning sales/marketing people to sales territories · Assigning accountants to accounts of the clients · Assigning machines to factory orders · Assigning contracts to bidders Characteristics of an Assignment Model (Hillier & Lieberman, 2015) · Each entity is to be assigned to exactly one (1) task · Each task is to be performed by exactly one entity Steps in Solving Assignment Model using Hungarian Method Hungarian Method (Flood’s Technique or Matrix Reduction Method)
· It is used to find minimum matches, in which the time of completion or cost of making all activities by a number of persons are minimized · It was first published by Harold W. Kuhn in 1955 · It was based on the earlier work of the two (2) Hungarian mathematicians: Dénes König and Jenö Egerváry
Sample Maximization Problem: · A department store has five (5) sections · They have five (5) employees available for service · The supervisor’s objective is to assign five (5) employees to five (5) sections in a way that will result in the highest profit
Sample Minimization Problem: · An electronics firm quality control records indicates that different number of defects on four (4) electronic components were produce by four (4) employees · The electronics firm objective is to create a set of assignments that will minimize the total number of defects produced by the firm
Unbalanced Assignment Model What is an Unbalanced Assignment Model? (Sirug, 2012) · As the name implies, this is a type of problem in which the number of entities to be assigned does not equal the number of tasks · To satisfy the one-to-one relationship for this, dummy rows or dummy columns are added Sample Minimization Problem: (Sirug, 2012) Determine the minimum combination on each row and column of Table below using Hungarian method.
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It may be noted that the assignment problem is a variation of transportation problem with two characteristics.(i)the cost matrix is a square matrix, and. (ii)
The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities
One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project). An
The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum.
Assignment Problem is a special type of linear programming problem where the ... Several problems of management have a structure identical with the
Give incremental due dates. Large comprehensive assignments due at the course finality leads to unfocused, or even plagiarized, writing. Break
Operation Research (IE 255320). ©Copyright ... Characteristics of Assignment Problem ... Algorithm for Assignment Problem: Hungarian Method. ○ Example:.
Constraints: Restrictions placed on the firm by the operating environment stated in linear relationships of the decision variables. Parameters: Numerical
Assumptions in Assignment Problem · Number of jobs is equal to the number of machines or persons. · Each man or machine is assigned only one job. · Each man or
What is an Assignment Model? (Sirug, 2012) It is a special case of transportation problem that requires pairing of various items to various receivers in