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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
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- 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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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research
Chapter: 12th business maths and statistics : chapter 10 : operations research.
Solution of assignment problems (Hungarian Method)
First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.
Step :1 Choose the least element in each row and subtract it from all the elements of that row.
Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.
Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.
Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.
Example 10.7
Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.
Here the number of rows and columns are equal.
∴ The given assignment problem is balanced. Now let us find the solution.
Step 1: Select a smallest element in each row and subtract this from all the elements in its row.
Look for atleast one zero in each row and each column.Otherwise go to step 2.
Step 2: Select the smallest element in each column and subtract this from all the elements in its column.
Since each row and column contains atleast one zero, assignments can be made.
Step 3 (Assignment):
Thus all the four assignments have been made. The optimal assignment schedule and total cost is
The optimal assignment (minimum) cost
Example 10.8
Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.
∴ The given assignment problem is balanced.
Now let us find the solution.
The cost matrix of the given assignment problem is
Column 3 contains no zero. Go to Step 2.
Thus all the five assignments have been made. The Optimal assignment schedule and total cost is
The optimal assignment (minimum) cost = ` 9
Example 10.9
Solve the following assignment problem.
Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is
Here only 3 tasks can be assigned to 3 men.
Step 1: is not necessary, since each row contains zero entry. Go to Step 2.
Step 3 (Assignment) :
Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is
The optimal assignment (minimum) cost = ₹ 35
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A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM
In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS. I have proposed two new methods Row Penalty Assignment Method (RPAM) and Column Penalty Assignment Method (CPAM) to obtain an IBFS of an UBAP. Also I have proposed a new method Non-basic Smallest Effectiveness Method (NBSEM) to test optimality of an IBFS. We can solve an assignment problem of maximization type using this new approach in opposite sense. By this new approach, we achieve the goal with less number of computations and steps. Further we illustrate the new approach by suitable examples. INTRODUCTION The assignment problem is a special case of the transportation problem where the resources are being allocated to the activities on a one-to-one basis. Thus, each resource (e.g. an employee, machine or time slot) is to be assigned uniquely to a particular activity (e.g. a task, site or event). In assignment problems, supply in each row represents the availability of a resource such as a man, machine, vehicle, product, salesman, etc. and demand in each column represents different activities to be performed such as jobs, routes, factories, areas, etc. for each of which only one man or vehicle or product or salesman respectively is required. Entries in the square being costs, times or distances. The assignment method is a special linear programming technique for solving problems like choosing the right man for the right job when more than one choice is possible and when each man can perform all of the jobs. The ultimate objective is to assign a number of tasks to an equal number of facilities at minimum cost (or maximum profit) or some other specific goal. Let there be 'm' resources and 'n' activities. Let c ij be the effectiveness (in terms of cost, profit, time, etc.) of assigning resource i to activity j (i = 1, 2, …., m; j = 1, 2,…., n). Let x ij = 0, if resource i is not assigned to activity j and x ij = 1, if resource i is assigned to activity j. Then the objective is to determine x ij 's that will optimize the total effectiveness (Z) satisfying all the resource constraints and activity constraints. 1. Mathematical Formulation Let number of rows = m and number of columns = n. If m = n then an AP is said to be BAP otherwise it is said to be UBAP. A) Case 1: If m < n then mathematically the UBAP can be stated as follows:
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IOP Publishing
Hussein Ali Hussein Al-Dallal Al-Saeedi
YMER Digital
Assignment model comes under the class of linear programming model which is the most used techniques of operations research, which looks alike with transportation model with an objective function of minimizing the time or cost of manufacturing the products by allocating one job to one machine or one machine to one job or one destination to one origin or one origin to one destination only. In this paper, we represent linear mathematical formulation of Assignment problem and solved using Lingo Software. Keyword: Resource Allocation, Optimization Problem, Lingo Software, Assignment problem.
archana pandey
Assignment problems arise in different situation where we have to find an optimal way to assign n-objects to mother objects in an injective fashion. The assignment problems are a well studied topic in combinatorial optimization. These problems find numerous application in production planning, telecommunication VLSI design, economic etc. The assignment problems is a special case of Transportation problem. Depending on the objective we want to optimize, we obtain the typical assignment problems. Assignment problem is an important subject discussed in real physical world we endeavor in this paper to introduce a new approach to assignment problem namely, matrix ones assignment method or MOA-method for solving wide range of problem. An example using matrix ones assignment methods and the existing Hungarian method have been solved and compared it graphically. Also some of the variations and some special cases in assignment problem and its applications have been discussed in the paper.
Industrial Engineering Journal
Shridhar Mhalsekar
IJAR Indexing
Assignment problems deal with the question how to assign n objects to m other objects in an injective fashion in the best possible way. An assignment problem is completely specified by its two components the assignments, which represent the underlying combinatorial structure, and the objective function to be optimized, which models \\\\\\\"the best possible way\\\\\\\". The assignment problem refers to another special class of linear programming problem where the objective is to assign a number of resources to an equal number of activities on a one to one basis so as to minimize total costs of performing the tasks at hand or maximize total profit of allocation. In this paper we introduce a new technique to solve assignment problems namely, Divide Row Minima and Subtract Column Minima .For the validity and comparison study we consider an example and solved by using our technique and the existing Hungarian (HA) and matrix ones assignment method(MOA) and compare optimum result shown graphically.
Ajit Pal Singh
Applied Mathematical Sciences
Anwar N Jasim
International Journal for Research in Applied Science & Engineering Technology (IJRASET)
IJRASET Publication
In this paper a new method is proposed for finding an optimal solution of a wide range of assignment problems, directly. A numerical illustration is established and the optimality of the result yielded by this method is also checked. The most attractive feature of this method is that it requires very simple arithmetical and logical calculations. The method is illustrated through an example.
American Journal of Operations Research
Mr Ebenezer Quayson
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5.2 ASSIGNMENT PROBLEM AND ITS SOLUTION An assignment problem may be considered as a special type of transportation problem in which the number of sources and destinations are equal. The capacity of each source as well as the requirement of each destination is taken as 1. In the case of an assignment problem, the given matrix must necessarily
formulate special linear programming problems using the assignment model solve assignment problems with the Hungarian method. 4.2 Introduction ... To illustrate how to set up a transportation problem we consider the following example; Example 4.1 A concrete company transports concrete from three plants, 1, 2 and 3, to three construction
Prasad A Y, Dept of CSE, ACSCE, B'lore-74. Page 33. Module 4: Transportation Problem and Assignment problem. This means that programmer 1 is assigned programme C, programmer 2 is assigned programme A, and so on. The minimum time taken in developing the programmes is = 80 + 80 + 100 + 90 = 350 min.
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
Problem 4. Job shop needs to assign 4 jobs to 4 workers. The cost of performing a job is a function of the skills of the workers. Table summarizes the cost of the assignments. Worker1 cannot do job3, and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method. Job. Worker.
Hungarian method for assignment problem Step 1. Subtract the entries of each row by the row minimum. Step 2. Subtract the entries of each column by the column minimum. Step 3. Make an assignment to the zero entries in the resulting matrix. A = M 17 10 15 17 18 M 6 10 20 12 5 M 14 19 12 11 15 M 7 16 21 18 6 M −10
8.1 Introduction. An assignment problem is a particular case of transportation problem in which a number of operations are to be assigned to an equal number of operators, where each operator performs only one operation. The objective is to minimize overall cost or to maximize the overall profit for a given assignment schedule.
Step 3. Cover all the zeros of the matrix with the minimum number of horizontal or vertical lines. Step 4. Since the minimal number of lines is 3, an optimal assignment of zeros is possible and we are finished. Since the total cost for this assignment is 0, it must be. Step 3.
Chapter 17 The Assignment Problem 301 These problems are all examples of problems which may be solved as as-signment problems. In this chapter we will derive an efficient algorithm for solving assignment problems, and then discuss several problems which may be solved using this algorithm. The assignment problem will then be described in terms ...
Chapter5 Thetransportationproblemandthe assignmentproblem Inthischapterweintroducethealgorithmsusedtosolvetwospecificlinearprob-lems ...
Several problems of management have a structure identical with the assignment problem. For example: Example I: A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost ... Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as ...
9. Do the same for some variants of assignment problems. 10. Give the name of an algorithm that can solve huge assignment problems that are well beyond the scope of Solver. Transportation problems were introduced in Section 3.5 and Section 3.6 did the same for assignment problems.
Lesson 20 :Solving Assignment problem Learning objectives: • Solve the assignment problem using Hungarian method. • Analyze special cases in assignment problems. Writing of an assignment problem as a Linear programming problem Example 1. Three men are to to be given 3 jobs and it is assumed that
Step 1. Determine the cost table from the given problem. If the no. of sources is equal to no. of destinations, go to step 3. 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.
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 ...
EXAMPLE OF ASSIGMENT PROBLEMS PROBLEM 1: Solve the following assignment problem shown in Table using Hungarian method. The matrix entries are processing time of each man in hours. 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I II III IV V 1 20 15 18 20 25 2 18 20 12 14 15 3 21 23 25 27 25
We outline below a generic example of such a Problem Solving assignment. Note that this exercise can be conducted individually or as a team. The assignment is based on the concept of "Wicked Problems" in society. A Wicked Problems is, by definition, a social or cultural problem that is difficult or impossible to solve for as many as four ...
The optimal assignment (minimum) cost = ` 9. Example 10.9. Solve the following assignment problem. Solution: Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is
the large economic burden, and the interconnected nature of these problems with other problems. Examples of Wicked Problems include poverty, education, the economy, pollution, and, of course, climate change. A 2015 article in Elephant Journal entitled, "Top 5 Wicked Problems the World Desperately Needs to Solve" lists and
Problem Set 2 (PDF) Problem Set 2 Solutions (PDF) ... assignment_turned_in Problem Sets with Solutions. grading Exams with Solutions. notes Lecture Notes. co_present Instructor Insights. Download Course. Over 2,500 courses & materials Freely sharing knowledge with learners and educators around the world.
A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM. Bhausaheb G Kore. In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS.
Bhausaheb G Kore. In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS. I have proposed two new methods Row Penalty Assignment Method (RPAM) and Column Penalty ...
7. PDF. Recently, Yadaiah and Haragopal published in the American Journal of Operations Research a new approach to solving the unbalanced assignment problem. They also provide a numerical example which they solve with their approach and get a cost of 1550 which they claim is optimum. This approach might be of interest; however, their approach ...