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Lesson 9. solution of assignment problem.
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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:
![assignment model questions pdf](https://www.engineeringenotes.com/wp-content/uploads/2017/03/clip_image002_thumb-74.jpg)
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Hungarian Method for Solving Assignment Problem:
The Hungarian method of assignment provides us with an efficient method of finding the optimal solution without having to make a-direct comparison of every solution. It works on the principle of reducing the given cost matrix to a matrix of opportunity costs.
Opportunity cost show the relative penalties associated with assigning resources to an activity as opposed to making the best or least cost assignment. If we can reduce the cost matrix to the extent of having at least one zero in each row and column, it will be possible to make optimal assignment.
The Hungarian method can be summarized in the following steps:
Step 1: Develop the Cost Table from the given Problem:
If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero.
Step 2: Find the Opportunity Cost Table:
(a) Locate the smallest element in each row of the given cost table and then subtract that from each element of that row, and
(b) In the reduced matrix obtained from 2 (a) locate the smallest element in each column and then subtract that from each element. Each row and column now have at least one zero value.
Step 3: Make Assignment in the Opportunity Cost Matrix:
The procedure of making assignment is as follows:
(a) Examine rows successively until a row with exactly one unmarked zero is obtained. Make an assignment single zero by making a square around it.
(b) For each zero value that becomes assigned, eliminate (Strike off) all other zeros in the same row and/ or column
(c) Repeat step 3 (a) and 3 (b) for each column also with exactly single zero value all that has not been assigned.
(d) If a row and/or column has two or more unmarked zeros and one cannot be chosen by inspection, then choose the assigned zero cell arbitrarily.
(e) Continue this process until all zeros in row column are either enclosed (Assigned) or struck off (x)
Step 4: Optimality Criterion:
If the member of assigned cells is equal to the numbers of rows column then it is optimal solution. The total cost associated with this solution is obtained by adding original cost figures in the occupied cells.
If a zero cell was chosen arbitrarily in step (3), there exists an alternative optimal solution. But if no optimal solution is found, then go to step (5).
Step 5: Revise the Opportunity Cost Table:
Draw a set of horizontal and vertical lines to cover all the zeros in the revised cost table obtained from step (3), by using the following procedure:
(a) For each row in which no assignment was made, mark a tick (√)
(b) Examine the marked rows. If any zero occurs in those columns, tick the respective rows that contain those assigned zeros.
(c) Repeat this process until no more rows or columns can be marked.
(d) Draw a straight line through each marked column and each unmarked row.
If a no of lines drawn is equal to the no of (or columns) the current solution is the optimal solution, otherwise go to step 6.
Step 6: Develop the New Revised Opportunity Cost Table:
(a) From among the cells not covered by any line, choose the smallest element, call this value K
(b) Subtract K from every element in the cell not covered by line.
(c) Add K to very element in the cell covered by the two lines, i.e., intersection of two lines.
(d) Elements in cells covered by one line remain unchanged.
Step 7: Repeat Step 3 to 6 Unlit an Optimal Solution is Obtained:
The flow chart of steps in the Hungarian method for solving an assignment problem is shown in following figures:
![logo assignment model questions pdf](https://www.engineeringenotes.com/wp-content/uploads/2017/03/clip_image005_thumb-18.jpg)
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Assignment Problem: Maximization
There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.
The Hungarian Method can also solve such assignment problems , as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss.
The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.
- Unbalanced Assignment Problem
- Multiple Optimal Solutions
Example: Maximization In An Assignment Problem
At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 30 | 37 | 40 | 28 | 40 |
2 | 40 | 24 | 27 | 21 | 36 |
3 | 40 | 32 | 33 | 30 | 35 |
4 | 25 | 38 | 40 | 36 | 36 |
5 | 29 | 62 | 41 | 34 | 39 |
How should the counters be assigned to persons so as to maximize the profit ?
Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.
On small screens, scroll horizontally to view full calculation
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 32 | 25 | 22 | 34 | 22 |
2 | 22 | 38 | 35 | 41 | 26 |
3 | 22 | 30 | 29 | 32 | 27 |
4 | 37 | 24 | 22 | 26 | 26 |
5 | 33 | 0 | 21 | 28 | 23 |
Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 10 | 3 | 8 | ||
2 | 16 | 13 | 15 | 4 | |
3 | 8 | 7 | 6 | 5 | |
4 | 15 | 2 | 4 | ||
5 | 33 | 21 | 24 | 23 |
Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.
Select the smallest element from all the uncovered elements, i.e., 4. Subtract this 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. Repeating step 3, we obtain a solution which is shown in the following table.
Final Table: Maximization Problem
Use Horizontal Scrollbar to View Full Table Calculation
Person | |||||
---|---|---|---|---|---|
Counter | A | B | C | D | E |
1 | 14 | 3 | 8 | ||
2 | 12 | 9 | 11 | ||
3 | 4 | 3 | 2 | 1 | |
4 | 19 | 2 | 4 | ||
5 | 37 | 21 | 24 | 23 |
The total cost of assignment = 1C + 2E + 3A + 4D + 5B
Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.
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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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/mWVDliU.jpg)
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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/bTg336I.jpg)
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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/xTfXnkt.jpg)
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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/IcAT59z.jpg)
Since each row and column contains atleast one zero, assignments can be made.
Step 3 (Assignment):
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/pbIKFmq.jpg)
Thus all the four assignments have been made. The optimal assignment schedule and total cost is
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/iAG3Efh.jpg)
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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/XjEMd93.jpg)
∴ The given assignment problem is balanced.
Now let us find the solution.
The cost matrix of the given assignment problem is
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/udWxK6z.jpg)
Column 3 contains no zero. Go to Step 2.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/xHaCVkG.jpg)
Thus all the five assignments have been made. The Optimal assignment schedule and total cost is
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/PgErQrY.jpg)
The optimal assignment (minimum) cost = ` 9
Example 10.9
Solve the following assignment problem.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/07kBJTh.jpg)
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
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/EROUf09.jpg)
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.
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/DyhURHX.jpg)
Step 3 (Assignment) :
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/ugeZxDF.jpg)
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
![assignment model questions pdf assignment model questions pdf](https://img.brainkart.com/imagebk40/gkDj91E.jpg)
The optimal assignment (minimum) cost = ₹ 35
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4.3.2 Mathematical model of a transportation problem Before we discuss the solution of transportation problems we will introduce the notation used to describe the transportation problem and show that it can be formulated as a linear programming problem. We use the following notation; x ij= the number of units to be distributed from
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.
e minimisation problem.3. The assignment problem wherein the number of rows is not equal to the number of columns is said t. be an unbalanced problem. Such a problem is handled by introducing dummy row(s) if the number of rows is less than the number of columns and dummy column(s) if the number of columns is le.
This information leads to the spreadsheet model shown in Figure 15.2. All the data provided in Table 15.5 are displayed in the following data cells: UnitCost (D5:G7), Supply (J12:J14), and Demand (D17:G17). The decisions on shipping quantities are given by the changing cells, ShippingQuantity (D12:G14).
THE ASSIGNMENT MODELS a special case of the transportation model is the assignment model. This model is appropriate in problems, which involve the assignment of resources to tasks (e.g assign n persons to n different tasks or jobs). Just as the special structure of the transportation model allows for solution
Jobs with costs of M are disallowed assignments. The problem is to find the minimum cost matching of machines to jobs. Fig 1 Matrix model of the assignment problem. The network model is in shown in Fig.2. It is very similar to the transportatio external flows are all +1 or -1. The only relevant parameter for the assignment model is arc cost
QUESTION TO ANSWER MCQ QUESTIONS WITH ANSWER K.BHARATHI,SCSVMV. ASSIGNMENT PROBLEM 2 / 55 ... to an equal number of activities . So as to minimize total cost or maximize total pro t of allocation. The problem of assignment arises because available resources such as men, machines etc. have varying degrees of e ciency for performing di erent ...
model file. Clearly we want to set up a general model to deal with this prob-lem. 3.2 An AMPL model for the transportation problem. Two fundamental sets of objects underlie the transportation problem: the sources or origins (mills, in our example) and the destinations (factories). Thus we begin the. AMPL. model with a declaration of these two sets:
Since the minimal number of lines is 3, an optimal assignment of zeros is possible and we are finished. 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 less than 4, we have to proceed to Step 5.
Step 1. For the original cost matrix, identify each row's minimum, and subtract it from all the entries of the row. Step 2. For the matrix resulting from step 1, identify each column's minimum, and sub tract it from all the entries of the column. Step 3. Identify the optimal assignment as the one associated with the zero elements of the ...
Chapter5 Thetransportationproblemandthe assignmentproblem. r 5The transportation problem and the assignment problemIn this chapter we introduce the algorithms used to solve two specific linear prob-le. nd the assignment problem.5.1 The transportation problemIn the application of linear programming techniques, the transportation problem w.
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
Unit-3 Assignement Model Notes & Practice Questions - Free download as PDF File (.pdf), Text File (.txt) or read online for free. The document discusses the assignment problem and provides an overview of the assignment procedure in 8 steps: 1. The objective is to either minimize cost or maximize profit. Jobs are assigned to facilities on a one-to-one basis.
The assignment problem can be solved by the following four methods: a) Complete enumeration method. b) Simplex Method. c) Transportation method. d) Hungarian method. 9.2.1 Complete enumeration method. In this method, a list of all possible assignments among the given resources and activities is prepared.
o each xij must equal 0 or 1.The assignment problem will be s. lved by the Hungarian method.Step 1: For the original cost matrix, identify each row's minimum, and subtract it fr. m all t. e entries of the row.Step 2. For the matrix resulting from step 1, identify each column's minimum, and subtract it from. ll the.
5.1 TRANSPORTATION MODELS. The transportation model is a special class of linear programming that deals with shipping a commodity from sources/origins (e.g Factory) to Destinations (e.g Warehouses). Each origin represents a source of supply for the commodity; each destination represents a point of Demand for the commodity.
Step 3. Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way: . For each row or column with a single zero value cell that has not be assigned or eliminated, box that zero value as an assigned cell.; For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.
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 ...
The task is to assign 1 job to 1 person so that the total number of hours are minimized. So, the first step in the assignment model would be to deduce all the numbers by the smallest number in the row. Hence, the smallest number becomes 0 and then we can target the zeroes to arrive at a conclusion.
The Hungarian Method can also solve such assignment problems, as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss. The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss ...
The optimal assignment (minimum) cost = ₹ 38. 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. Solution: Here the number of rows and columns are equal. ∴ The given assignment problem is balanced. Now let us find the solution.
1. Row D has only one "0", so make assignment to it (D4). And cross all other 0 in the corresponding column. 2. After Assignment of D on machine 4, now Row B and C both have only one Zero. So assignment are made to these rows ( C1 and B2) and zeros in columns are crossed. 3. Finally only one Zero is left in Row A where Machine three has a Zero
assignment model typical question - Free download as PDF File (.pdf), Text File (.txt) or read online for free.