Assignment Problem: Meaning, Methods and Variations | Operations Research

the assignment problem is said to be unbalanced if mcq

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:

the assignment problem is said to be unbalanced if mcq

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.

the assignment problem is said to be unbalanced if mcq

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.

the assignment problem is said to be unbalanced if mcq

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.

the assignment problem is said to be unbalanced if mcq

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

the assignment problem is said to be unbalanced if mcq

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

the assignment problem is said to be unbalanced if mcq

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 assignment problem is said to be unbalanced if mcq

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

the assignment problem is said to be unbalanced if mcq

Column 3 contains no zero. Go to Step 2.

the assignment problem is said to be unbalanced if mcq

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

the assignment problem is said to be unbalanced if mcq

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

the assignment problem is said to be unbalanced if mcq

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 assignment problem is said to be unbalanced if mcq

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.

the assignment problem is said to be unbalanced if mcq

Step 3 (Assignment) :

the assignment problem is said to be unbalanced if mcq

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 assignment problem is said to be unbalanced if mcq

The optimal assignment (minimum) cost = ₹ 35

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OPERATIONS RESEARCH

Lesson 9. solution of assignment problem.

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UNBALANCED ASSIGNMENT PROBLEM

Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To make unbalanced assignment problem, a balanced one, a dummy facility(s) or a dummy job(s) (as the case may be) is introduced with zero cost or time.

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

hungarian1

  • 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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MBA Notes

Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

Table of Contents

Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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Operations Research

1 Operations Research-An Overview

  • History of O.R.
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2 Linear Programming: Formulation and Graphical Method

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  • Two Phase and M-method
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  • Sensitivity Analysis
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4 Transportation Problem

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  • Modified Distribution Method
  • Stepping Stone Method
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  • Degenerate Transportation Problem
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5 Assignment Problem

  • Solution of the Assignment Problem
  • Unbalanced Assignment Problem
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6 Application of Excel Solver to Solve LPP

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7 Goal Programming

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Operations Research

141. Which of the following is correct?

  • Re-order quantity in a fixed order-interval system equals EOQ
  • Review period of the item is always kept higher than its lead time
  • Re-order level of an item is always more than its minimum stock
  • Buffer stock is the total stock kept to meet the demand during lead time

142. The group replacement policy is suitable for identical low cost items which are likely to ______________.

  • fail suddenly
  • fail completely and suddenly
  • fail over a period of time
  • be progressive and retrogressive

143. Identify the correct statement

  • an assignment problem may require the introduction of both dummy row and dummy column
  • an assignment problem with m rows and n columns will involves a total of m x n possible assignments
  • an unbalanced assignment is one where the number of rows is more than, or less than the number of columns
  • balancing any unbalanced assignment problem involves adding one dummy row or column

144. A game is said to be fair if ______________.

  • lower and upper values are zero
  • only lower value to be zero
  • only upper value to be zero
  • lower and upper values are not equal to zero

145. Which of the following is not a part of holding (or carrying) costs?

  • Rent for storage space
  • Extra expenses for an overnight express mail.
  • Spoilage costs
  • Electricity and heat for the buildings

146. The area bounded by all the given constraints is called ______________.

  • feasible region
  • basic solution
  • non feasible region
  • optimum basic feasible solution

147. When D=18000, holding cost=Rs.1.20, set-up cost=Rs.400 ,EOQ = ______________

148. Given arrival rate = 15/hr, service rate = 20/hr, the value of traffic intensity is ______________.

149. An activity is critical if its ______________ float is zero

  • independent
  • interference

150. ______________ is employed in construction and business problems

  • Replacement

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The assignment problem is said to be unbalanced if

a.number of rows is greater than number of columns. b.number of rows is lesser than number of columns. c.number of rows is equal to number of columns. d.both 1 and 2.

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The assignment problem is said to be unbalanced if ______________.

Correct answer: (D) both 1 and 2.

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  2. The Assignment Problem Lecture 5: Unbalanced and Maximization Problems

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  3. An unbalanced assignment problem

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  4. Lecture 19 Assignment problem : Unbalanced and maximal Assignment Problems

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  5. EASY STEPS TO SOLVE UNBALANCED AND CONSTRAINED ASSIGNMENT PROBLEM PART 1

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COMMENTS

  1. MCQ Assignment

    a. only minimum cost. b. only maximum profit. c. minimum cost or maximum profit. d. assign the jobs. The assignment problem is said to be unbalanced if _________. a. number of rows is greater than number of columns. b.

  2. Solutions for Chapter 7: Assignment Problem and Sequencing

    MCQ Online Mock Tests 99. Important Solutions 3958. Concept Notes & Videos 145. ... An assignment problem is said to be unbalanced when _____. VIEW SOLUTION. Miscellaneous Exercise 7 | Q 2.02 | Page 127 ... with the cost elements as _____ is added to the matrix of an unbalanced assignment problem to convert into a square matrix. VIEW SOLUTION ...

  3. Solved 1. The assignment problem is said to be unbalanced if

    See Answer Question: 1. The assignment problem is said to be unbalanced if a. Number of rows is greater than number of columns b. Number of rows is less than number of columns c. Number of rows is equal to number of columns d. None of these Ans: 2. For finding Initial basic solution to transportation problem Method is Used a. Simplex method b.

  4. Assignment Problem: Meaning, Methods and Variations

    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.

  5. Solution of assignment problems (Hungarian Method)

    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. Here only 3 tasks can be assigned to 3 men.

  6. Business Mathematics Multiple choice Questions and Answers. Page 9

    Multiple choice Questions on Business Mathematics. Practice for BBA or MBA exams using these MCQ. Page 9. ... Transportation problem is said to be unbalanced if _____. ... The assignment problem is said to be balanced if _____.

  7. Assignment problem

    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. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. [1] If the total cost of the assignment for all ...

  8. ES-3: Lesson 9. SOLUTION OF ASSIGNMENT PROBLEM

    Mark (√) in the rows in which assignments has not been made. 2. Mark column with (√) which have zeros in the marked rows. 3. Mark rows with (√) which contains assignment in the marked column. 4. Repeat 2 and 3 until the chain of marking is completed. 5. Draw straight lines through marked columns.

  9. Unbalanced Assignment Problem

    Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To make unbalanced assignment problem, a balanced one, a dummy facility (s) or a dummy job (s) (as the case may be) is introduced with zero cost or time.

  10. The assignment problem is said to be unbalance if

    Options Number of rows is greater than number of columns Number of rows is lesser than number of columns Number of rows is equal to number of columns Both (a) and (b) Advertisement Solution Both (a) and (b). Concept: Assignment Problem Is there an error in this question or solution? Q 1.09 Q 1.08 Q 1.1

  11. PDF ASSIGNMENT PROBLEM

    EXAMPLE OF ASSIGMENT PROBLEMS QUESTION TO ANSWER MCQ QUESTIONS WITH ANSWER K.BHARATHI,SCSVMV. ASSIGNMENT PROBLEM 2 / 55. ... columns, if it is so, the assignment problem is said to be balanced. Then proceed to step 1. If it is not balanced, then it should be balanced before applying the algorithm.

  12. Hungarian Algorithm for Assignment Problem

    The Hungarian algorithm, aka Munkres assignment algorithm, utilizes the following theorem for polynomial runtime complexity ( worst case O (n3)) and guaranteed optimality: If a number is added to or subtracted from all of the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an ...

  13. Fill in the blank : An assignment problem is said to be unbalanced when

    Solution An assignment problem is said to be unbalanced when number of rows is not equal to the number of columns. Concept: Special Cases of Assignment Problem Is there an error in this question or solution? Q 2.01 Q 1.14 Q 2.02 Chapter 7: Assignment Problem and Sequencing - Miscellaneous Exercise 7 [Page 127] APPEARS IN

  14. Operations Research Multiple choice Questions and Answers. Page 13

    Multiple choice Questions on Operations Research. Practice for BBA or MBA exams using these MCQ. Page 13. ... In an assignment problem involving 5 workers and 5 jobs, total number of assignments possible are _____. 5; 10; 15; 20; ... unbalanced. 130. All equality constraints can be replaced equivalently by _____ inequalities. 1; 2; 3; 4;

  15. Unbalanced Assignment Problem: Definition, Formulation, and Solution

    The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has ...

  16. Operations Research Multiple choice Questions and Answers. Page 15

    an assignment problem with m rows and n columns will involves a total of m x n possible assignments; an unbalanced assignment is one where the number of rows is more than, or less than the number of columns; balancing any unbalanced assignment problem involves adding one dummy row or column; View answer

  17. The assignment problem is said to be unbalanced if

    a.number of rows is greater than number of columns. b.number of rows is lesser than number of columns. c.number of rows is equal to number of columns. d.both 1 and 2.

  18. FBA MCQ Flashcards

    C. 3. In Index method of solving assignment problem (a) The whole matrix is divided by smallest element (b) The smallest element is subtracted from whole matrix (c) Each row or column is divided by smallest element in that particular row or column (d) The whole matrix is multiplied by - 1.

  19. The assignment problem is said to be unbalanced if

    The assignment problem is said to be unbalanced if ______________. number of rows is greater than number of columns. number of rows is lesser than number of columns. number of rows is equal to number of columns. both 1 and 2. Related MCQs ? The objective of the transportation problem which is to be maximized is to ?

  20. PDF Solving The Assignment Problems Directly Without Any Iterations

    2-The Unbalanced Assignment problems is achieved when the number of resourses differs or not equal to the number of activities or tasks. in this research I propose a new method which gives the optimal solution directly without any iteration Contrary to previous methods such as Hungarian Method which begins with an initial solution and then ...

  21. Chapter 6 (multiple choice) Flashcards

    a. total supply must equal total demand in the transportation problem. b. the number of origins must equal the number of destinations in the transportation problem. c. each supply and demand value is 1 in the assignment problem. d. there are many differences between the transportation and assignment problem.