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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:

• 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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How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

Index     Assignment problem     Hungarian algorithm     Solve online    

The Hungarian algorithm: An example

We consider an example where four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker. The matrix below shows the cost of assigning a certain worker to a certain job. The objective is to minimize the total cost of the assignment.

Below we will explain the Hungarian algorithm using this example. Note that a general description of the algorithm can be found here .

Step 1: Subtract row minima

We start with subtracting the row minimum from each row. The smallest element in the first row is, for example, 69. Therefore, we substract 69 from each element in the first row. The resulting matrix is:

Step 2: Subtract column minima

Similarly, we subtract the column minimum from each column, giving the following matrix:

Step 3: Cover all zeros with a minimum number of lines

We will now determine the minimum number of lines (horizontal or vertical) that are required to cover all zeros in the matrix. All zeros can be covered using 3 lines:

Step 4: Create additional zeros

First, we find that the smallest uncovered number is 6. We subtract this number from all uncovered elements and add it to all elements that are covered twice. This results in the following matrix:

Now we return to Step 3.

Again, We determine the minimum number of lines required to cover all zeros in the matrix. Now there are 4 lines required:

Because the number of lines required (4) equals the size of the matrix ( n =4), an optimal assignment exists among the zeros in the matrix. Therefore, the algorithm stops.

The optimal assignment

The following zeros cover an optimal assignment:

This corresponds to the following optimal assignment in the original cost matrix:

Thus, worker 1 should perform job 3, worker 2 job 2, worker 3 job 1, and worker 4 should perform job 4. The total cost of this optimal assignment is to 69 + 37 + 11 + 23 = 140.

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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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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.

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

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.

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

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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• Business Essentials

Assignment Method: Examples of How Resources Are Allocated

What Is the Assignment Method?

The assignment method is a way of allocating organizational resources in which each resource is assigned to a particular task. The resource could be monetary, personnel , or technological.

Understanding the Assignment Method

The assignment method is used to determine what resources are assigned to which department, machine, or center of operation in the production process. The goal is to assign resources in such a way to enhance production efficiency, control costs, and maximize profits.

The assignment method has various applications in maximizing resources, including:

• Allocating the proper number of employees to a machine or task
• Allocating a machine or a manufacturing plant and the number of jobs that a given machine or factory can produce
• Assigning a number of salespersons to a given territory or territories
• Assigning new computers, laptops, and other expensive high-tech devices to the areas that need them the most while lower priority departments would get the older models

Companies can make budgeting decisions using the assignment method since it can help determine the amount of capital or money needed for each area of the company. Allocating money or resources can be done by analyzing the past performance of an employee, project, or department to determine the most efficient approach.

Regardless of the resource being allocated or the task to be accomplished, the goal is to assign resources to maximize the profit produced by the task or project.

Example of Assignment Method

A bank is allocating its sales force to grow its mortgage lending business. The bank has over 50 branches in New York but only ten in Chicago. Each branch has a staff that is used to bring in new clients.

The bank's management team decides to perform an analysis using the assignment method to determine where their newly-hired salespeople should be allocated. Given the past performance results in the Chicago area, the bank has produced fewer new clients than in New York. The fewer new clients are the result of having a small market presence in Chicago.

As a result, the management decides to allocate the new hires to the New York region, where it has a greater market share to maximize new client growth and, ultimately, revenue.

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> > Assignment Problem example (Using Hungarian method) ( ) ) 2. Algorithm & Example-1 If number of rows is not equal to number of columns, then add dummy rows or columns with cost 0, to make it a square matrix. a. Identify the minimum element in each row and subtract it from each element of that row. b. Identify the minimum element in each column and subtract it from every element of that column. Make assignment in the opporunity cost table a. Identify rows with exactly one unmarked 0. Make an assignmment to this single 0 by make a square ( [0] ) around it and cross off all other 0 in the same column. b. Identify columns with exactly one unmarked 0. Make an assignmment to this single 0 by make a square ( [0] ) around it and cross off all other 0 in the same rows. c. If a row and/or column has two or more unmarked 0 and one cannot be chosen by inspection, then choose the cell arbitarily. d. Continue this process until all 0 in rows/columns are either assigned or cross off( (a) If the number of assigned cells = the number of rows, then an optimal assignment is found and In case you have chosen a 0 cell arbitrarily, then there may be an alternate optimal solution exists. (b) If optimal solution is not optimal, then goto Step-5. Draw a set of horizontal and vertical lines to cover all the 0 a. Tick(✓) mark all the rows in which no assigned 0. b. Examine Tick(✓) marked rows, If any 0 cell occurs in that row, then tick(✓) mark that column. c. Examine Tick(✓) marked columns, If any assigned 0 exists in that columns, then tick(✓) mark that row. d. Repeat this process until no more rows or columns can be marked. e. Draw a straight line for each unmarked rows and marked columns. f. If the number of lines is equal to the number of rows then the current solution is the optimal, otherwise goto step-6 Develop the new revised opportunity cost table a. Select the minimum element, say k, from the cells not covered by any line, b. Subtract k from each element not covered by a line. c. Add k to each intersection element of two lines. Repeat steps 3 to 6 until an optimal solution is arrived. \ I II III IV V A 10 5 13 15 16 B 3 9 18 13 6 C 10 7 2 2 2 D 7 11 9 7 12 E 7 9 10 4 12     `I`   `II`   `III`   `IV`   `V`       `A`   `B`   `C`   `D`   `E`          `I`   `II`   `III`   `IV`   `V`       `A`   `5=10-5`  `0=5-5`  `8=13-5`  `10=15-5`  `11=16-5`  Minimum element of `1^(st)` row  `B`   `0=3-3`  `6=9-3`  `15=18-3`  `10=13-3`  `3=6-3`  Minimum element of `2^(nd)` row  `C`   `8=10-2`  `5=7-2`  `0=2-2`  `0=2-2`  `0=2-2`  Minimum element of `3^(rd)` row  `D`   `0=7-7`  `4=11-7`  `2=9-7`  `0=7-7`  `5=12-7`  Minimum element of `4^(th)` row  `E`   `3=7-4`  `5=9-4`  `6=10-4`  `0=4-4`  `8=12-4`  Minimum element of `5^(th)` row         `I`   `II`   `III`   `IV`   `V`       `A`   `5=5-0`  `0=0-0`  `8=8-0`  `10=10-0`  `11=11-0`  `B`   `0=0-0`  `6=6-0`  `15=15-0`  `10=10-0`  `3=3-0`  `C`   `8=8-0`  `5=5-0`  `0=0-0`  `0=0-0`  `0=0-0`  `D`   `0=0-0`  `4=4-0`  `2=2-0`  `0=0-0`  `5=5-0`  `E`   `3=3-0`  `5=5-0`  `6=6-0`  `0=0-0`  `8=8-0`      Minimum element of `1^(st)` column  Minimum element of `2^(nd)` column  Minimum element of `3^(rd)` column  Minimum element of `4^(th)` column  Minimum element of `5^(th)` column     `I`   `II`   `III`   `IV`   `V`       `A`   (1) Rowwise cell `(A,II)` is assigned  `B`   (2) Rowwise cell `(B,I)` is assigned so columnwise cell `(D,I)` crossed off.  `C`   (4) Columnwise cell `(C,III)` is assigned so rowwise cell `(C,V)` crossed off.  Columnwise `(C,IV)` crossed off because (3) Rowwise cell `(D,IV)` is assigned  Rowwise `(C,V)` crossed off because (4) Columnwise cell `(C,III)` is assigned  `D`   Columnwise `(D,I)` crossed off because (2) Rowwise cell `(B,I)` is assigned  (3) Rowwise cell `(D,IV)` is assigned so columnwise cell `(C,IV)`,`(E,IV)` crossed off.  `E`   Columnwise `(E,IV)` crossed off because (3) Rowwise cell `(D,IV)` is assigned         `I`   `II`   `III`   `IV`   `V`       `A`   `B`   (5) Mark(✓) row `B` since column `I` has an assignment in this row `B`.  `C`   `D`   (3) Mark(✓) row `D` since column `IV` has an assignment in this row `D`.  `E`   (1) Mark(✓) row `E` since it has no assignment      (4) Mark(✓) column `I` since row `D` has 0 in this column  (2) Mark(✓) column `IV` since row `E` has 0 in this column     `I`   `II`   `III`   `IV`   `V`       `A`   `7=5+2` intersection cell of two lines  cell covered by a line  cell covered by a line  `12=10+2` intersection cell of two lines  cell covered by a line  `B`   cell covered by a line  `4=6-2` cell not covered by a line  `13=15-2` cell not covered by a line  cell covered by a line  `1=3-2` cell not covered by a line  `C`   `10=8+2` intersection cell of two lines  cell covered by a line  cell covered by a line  `2=0+2` intersection cell of two lines  cell covered by a line  `D`   cell covered by a line  `2=4-2` cell not covered by a line  `0=2-2` cell not covered by a line  cell covered by a line  `3=5-2` cell not covered by a line  `E`   cell covered by a line  `3=5-2` cell not covered by a line  `4=6-2` cell not covered by a line  cell covered by a line  `6=8-2` cell not covered by a line         `I`   `II`   `III`   `IV`   `V`       `A`   (1) Rowwise cell `(A,II)` is assigned  `B`   (2) Rowwise cell `(B,I)` is assigned so columnwise cell `(D,I)` crossed off.  `C`   Rowwise `(C,III)` crossed off because (4) Columnwise cell `(C,V)` is assigned  (4) Columnwise cell `(C,V)` is assigned so rowwise cell `(C,III)` crossed off.  `D`   Columnwise `(D,I)` crossed off because (2) Rowwise cell `(B,I)` is assigned  (5) Rowwise cell `(D,III)` is assigned  Columnwise `(D,IV)` crossed off because (3) Rowwise cell `(E,IV)` is assigned  `E`   (3) Rowwise cell `(E,IV)` is assigned so columnwise cell `(D,IV)` crossed off.         `I`   `II`   `III`   `IV`   `V`       `A`   Original cost 10  Original cost 5  Original cost 13  Original cost 15  Original cost 16  `B`   Original cost 3  Original cost 9  Original cost 18  Original cost 13  Original cost 6  `C`   Original cost 10  Original cost 7  Original cost 2  Original cost 2  Original cost 2  `D`   Original cost 7  Original cost 11  Original cost 9  Original cost 7  Original cost 12  `E`   Original cost 7  Original cost 9  Original cost 10  Original cost 4  Original cost 12     Work Job Cost `A` `II` `B` `I` `C` `V` `D` `III` `E` `IV` Total 23 Your Article Library Assignment problem in linear programming : introduction and assignment model. ADVERTISEMENTS: Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems. In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem. 1. Assignment Model : Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized. In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns. Mathematical Formulation: Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem. Suppose x jj is a variable which is defined as 1 if the i th job is assigned to j th machine or facility 0 if the i th job is not assigned to j th machine or facility. Now as the problem forms one to one basis or one job is to be assigned to one facility or machine. The total assignment cost will be given by The above definition can be developed into mathematical model as follows: Determine x ij > 0 (i, j = 1,2, 3…n) in order to Subjected to constraints and x ij is either zero or one. Method to solve Problem (Hungarian Technique): Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem, 1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table. 2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table. 3. Now, the assignments are made for the reduced table in following manner. (i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row. (ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column. (iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4. 4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted: (iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns. (iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked. (v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them. 5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6. 6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments. 7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns. Related Articles: Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase Linear Programming: Applications, Definitions and Problems No comments yet. Leave a reply click here to cancel reply.. You must be logged in to post a comment. Jump to menu Student Home Accept your offer How to enrol Student ID card Set up your IT Orientation Week Fees & payment Academic calendar Special consideration Transcripts The Nucleus: Student Hub Referencing Essay writing Learning abroad & exchange Professional development & UNSW Advantage Employability Financial assistance International students Equitable learning Postgraduate research Health Service Events & activities Emergencies Volunteering Clubs and societies Accommodation Health services Sport and gym Arc student organisation Security on campus Maps of campus Careers portal Change password Answering Assignment Questions In order to decide how to answer an essay question, you need to identify what the question requires in terms of content and genre. This guide outlines some methods to help you analyse essay questions. Analyse the question using key words Assignment questions can be broken down into parts so that you can better understand what you are being asked to do. It is important to identify key words and phrases in the topic. What are key words? Key words are the words in an assignment question that tell you the approaches to take when you answer. Make sure you understand the meaning of key words in an essay question, especially t ask words . As Task words are verbs that direct you and tell you how to go about answering a question, understanding the meaning helps you know exactly what you to do. Content words tell you what the topic area(s) of your assignment are and take you halfway towards narrowing down your material and selecting your answer. Content words help you to focus your research and reading on the correct area. Limiting words make a broad topic workable. They focus the topic area further by indicating aspects you should narrowly concentrate on. If you're not sure about any aspect of the question, ask your tutor/lecturer for clarification. Never start any assignment until you know and understand exactly what you are being asked to do. How to use key words Look for the keywords in your essay question. Underline them. Spend a little time working out what they mean. Use the Glossary of task words to help you. Example Question Computers have had a significant impact on education in the 20th century. Discuss the changes they have made. DISCUSS. Look up the meaning in the glossary of task words to find out what it means. (See Glossary of task words ) Content Words EDUCATION, COMPUTERS. Content words help you to direct your research and reading towards the correct area(s), in this case on computers and on education. Limiting Words CHANGES, SIGNIFICANT IMPACT, 20TH CENTURY. Limiting words further define the topic area and indicate aspects you should narrowly concentrate on. For example, in this question, do not just write about computers in education, Discuss the SIGNIFICANT IMPACT they have had and the CHANGES computers have made to education during a certain time: the 20TH CENTURY.  See next: Implied or complex questions Essay and assignment writing guide. Essay writing basics Essay and assignment planning Complex assignment questions Glossary of task words Editing checklist Writing a critical review Annotated bibliography Reflective writing ^ More support Understanding your assignment questions: A short guide Introduction Breaking down an assignment question Ways to get started, how do you narrow down a broad or general essay question, parts of a question, specific vs general essay questions. Further reading and references A-Z of Other Guides This link opens in a new window Academic Skills Gateway This link opens in a new window Book an Academic Skills Team Appointment This link opens in a new window Before you attempt to answer an assignment question, you need to make sure you understand what it is asking. This includes the subject matter, but also the way in which you are required to write. Different questions may ask you to discuss, outline, evaluate...and many more. The task words are a key part of the question. Once you have broken down and understood your assignment question, you can start to jot down your ideas, organise your research, and figure out exactly what point you want to argue in your essay. Here is something to try if you are struggling to get going with responding to the assignment question: Try to come up with a one word answer to the question ('yes,' 'no',' maybe'- or perhaps two words: 'not quite'; ''only sometimes'); Then expand the one-word answer into a sentence summarising your reason for saying that; Then expand that sentence into three sentences. This could be the beginning of your essay plan. Choose one or two key aspects of the topic to focus your argument around. Focus on a few examples rather than trying to cover everything that falls under that topic. Decide on a standpoint you want to argue (this applies to specific essay questions too). Make sure your introduction explains your chosen focus aim and argument.  Directive or task words : Tell you exactly what to do e.g., discuss, argue etc. Subject matter : Specifically what you should be writing about.  Limiting words : Parts of the question that may narrow or alter the focus of your answer.  Example : To what extent can the novel White Teeth by Zadie Smith be read differently in the light of the 9/11 Terrorist Attack? To what extent: This indicates you will need to explore both sides of the topic in a critical way and reach a decision Be read differently: This limiting phrase indicates that you will not be writing everything you know about White Teeth and 9/11. You will be focussing on whether or not the terrorist attack alters our reading of the novel. Every point you make should contribute to this. White Teeth: You will need to focus on this novel 9/11 Terrorist Attack: You will also need to write about this event in relation to the novel Some essay questions may have a narrow focus e.g., 'To what extent can it be argued that Byron and Keats are second generation Romantic poets?'.  While other may be quite broad e.g., ' Evaluate the effect of landscape on the expansion of the town'. The first example indicates exactly which poets to focus on, and which aspect of their work to explore. The second example is much broader: it doesn't specify which features of landscape, or which towns should be analysed.  Even if the essay question is broad, your answer should have a clear and specific focus. Therefore, you need to choose an area of the topic to concentrate on. If answering the second of the two questions above, you would not need to write about the impact if every type of landscape on every town in the world. It is normally better to write a lot about a little, rather than a little about a lot.  It is also important to note that, although the specific essay question tells you which poets to focus on and which aspect of their work to discuss, it does not dictate which way you have to argue. You are still free to choose your own standpoint (based on evidence) as to whether or not Byron and Keats can be seen as second generation Romantic poets.  << Previous: Introduction Next: Task words >> Last Updated: Aug 5, 2024 4:28 PM URL: https://libguides.bham.ac.uk/asc/understandingassignments Pardon Our Interruption As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen: You've disabled JavaScript in your web browser. You're a power user moving through this website with super-human speed. You've disabled cookies in your web browser. A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article . To regain access, please make sure that cookies and JavaScript are enabled before reloading the page. trending now in Lifestyle These airports make passengers walk the longest distances... Common condition linked to a 36% higher risk for Alzheimer's... The terrifying reason why you should never run your dishwasher at... I make almost $1M a year from my dream restaurant — but I... I was stunned to find this petty fee on my $270 restaurant bill... We spent $118K on a 9-month world cruise to see 65 countries:... I'm a Gen Z travel influencer who's been to 190 countries —... My dad murdered my mom then killed himself -- now I'm turning it... Elon musk’s favorite job interview question is a secret method to catch liars. You’ll get the “X” if you fail this job interview question. Elon Musk, Space X founder and Tesla CEO revealed his favorite, time-tested job interview question that catches liars. During the 2017 World Government Summit, Musk admitted he asked all job applicants, “Tell me about some of the most difficult problems you worked on and how you solved them.” Musks believes the question can eliminate dishonest candidates, which is proven by “Asymmetric Information Management (AIM).” The AIM approach “enhances verbal lie-detection by encouraging truth-tellers (but not liars) to be forthcoming with information,” explains Applied Cognitive Psychology . There are several ways to spot a liar, according to a study published in the Journal of Applied Research in Memory and Cognition in December 2020. If an applicant answers Musk’s question with a detailed response, they’re likelier to tell the truth, unlike someone who gives a vague answer. “Small details are the lifeblood of forensic investigations and can provide investigators with facts to check and witnesses to question,” Cody Porter, one of the study’s authors, wrote. Liars will provide little to no detail to “conceal their guilt,” which could be perceived as strategically withholding information when the AIM method is introduced. “Their assumption here is that providing more information will make it easier for the investigator to detect their lie, so instead, they provide less information,” Ported added. Although Musk uses the AIM method to cut down applicants, he seeks employees who embody “evidence of exceptional ability.” To ensure they match up with their resume, Musk conducts interviews that allow him to detect if the candidate is really who they say they are. “If there’s a track record of exceptional achievement, then it’s likely that that will continue into the future,” Porter said.  Challenges in detecting ecological interactions using sedimentary ancient DNA data Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Fiona Margaret Callahan For correspondence: [email protected] Info/History Supplementary material Preview PDF With increasing availability of ancient and modern environmental DNA technology, whole-community species occurrence and abundance data over time and space is becoming more available. Sedimentary ancient DNA data can be used to infer associations between species, which can generate hypotheses about biotic interactions, a key part of ecosystem function and biodiversity science. Here, we have developed a realistic simulation to evaluate five common methods from different fields for this type of inference. We find that across all methods tested, false discovery rates of inter-species associations are high under realistic simulation conditions. Additionally, we find that with sample sizes that are currently realistic for this type of data, models are typically unable to detect interactions better than random assignment of associations. We also find that at larger sample sizes, information about species abundance improves performance of these models. Different methods perform differentially well depending on the number of taxa in the dataset. Some methods (SPIEC-EASI, SparCC) assume that there are large numbers of taxa in the dataset, and we find that SPIEC-EASI is highly sensitive to this assumption while SparCC is not. We find that for small numbers of species, no method consistently outperforms logistic and linear regression, indicating a need for further testing and methods development. Competing Interest Statement The authors have declared no competing interest. View the discussion thread. Supplementary Material Thank you for your interest in spreading the word about bioRxiv. NOTE: Your email address is requested solely to identify you as the sender of this article. 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TEACHING METHODOLOGY Scientific Method COMMENTS Solve the assignment problem online Solve an assignment problem online. Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given. Fill in the cost matrix (random cost matrix): Assignment Problem: Meaning, Methods and Variations 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 ... Assignment Method Frequently Asked Questions (FAQs) 1. How does Johnson's rule differ from the assignment method? Johnson's rule is an operations research method that aims to estimate the optimal sequence of jobs in two work centers to reduce makespan. It optimizes the overall efficiency of the process. In contrast, the assignment method is useful for ... PDF Unit 4: ASSIGNMENT PROBLEM 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. Hungarian Algorithm for Assignment Problem 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. Hungarian Method Examples, Assignment Problem Example 1: Hungarian Method. The Funny Toys Company 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 in such a way that the total cost of assignment is minimum. Job. How to Solve the Assignment Problem: A Complete Guide The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment. Solution of the Assignment Problem using the Hungarian Method. The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. PDF The Assignment Problem: An Example 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 An Assignment Problem solved using the Hungarian Algorithm The matrix below shows the cost of assigning a certain worker to a certain job. The objective is to minimize the total cost of the assignment. Below we will explain the Hungarian algorithm using this example. Note that a general description of the algorithm can be found here. Step 1: Subtract row minima. PDF The Assignment Problem and the Hungarian Method Step 3. Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used. Step 4. Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished. Solution of assignment problems (Hungarian Method) 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. Assignment Problem, Maximization Example, Hungarian Method 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 ... Assignment methods: matching biological questions with appropriate Assignment methods, which use genetic information to ascertain population membership of individuals or groups of individuals, have been used in recent years to study a wide range of evolutionary and ecological processes. In applied studies, the first step of articulating the biological question (s) to be addressed should be followed by ... Assignment Model in Operation Research Assignment model is a technique to assign multiple jobs to different people or machines. Understand the types and example of assignment model. ... For this, a Hungarian mathematician developed a method called the assignment model, which can be used to solve such problems. (Note that this method is useful when the number of sources equals the ... PDF UNIT 5 ASSIGNMENT PROBLEMS Assignment Problems 7 Hungarian Method of Solving an Assignment Problem The steps for obtaining an optimal solution of an assignment problem are as follows: 1. Check whether the given matrix is square. If not, make it square by adding a suitable number of dummy rows (or columns) with 0 cost/time elements. 2. Assignment Method: Examples of How Resources Are Allocated Assignment Method: A method of allocating organizational resources. The assignment method is used to determine what resources are assigned to which department, machine or center of operation in ... Assignment problem using Hungarian method Algorithm & Example-1 Algorithm & Example-1. Algorithm. Hungarian Method Steps (Rule) Step-1: If number of rows is not equal to number of columns, then add dummy rows or columns with cost 0, to make it a square matrix. Step-2: a. Identify the minimum element in each row and subtract it from each element of that row. Quiz & Worksheet The use of the Hungarian Algorithm for the solving of assignment problems is the subject of this quiz and worksheet combo. Quiz questions ask about why the Hungarian Algorithm is used, as well as ... Assignment Problem in Linear Programming : Introduction and Assignment Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by ... Answering Assignment Questions This guide outlines some methods to help you analyse essay questions. Analyse the question using key words. ... Key words are the words in an assignment question that tell you the approaches to take when you answer. Make sure you understand the meaning of key words in an essay question, especially task words. As Task words are verbs that direct ... PDF Unpacking an Assignment Question & Planning for an Assignment Analysing an assignment question For every assignment you need to : • Understand the question • Brainstorm ideas • Read enough relevant, reliable information to answer the question properly. • Structure your assignment • Present your main ideas or arguments and • Support your main ideas or arguments • Edit your writing for coherence and unity • Proofread for grammar ... Understanding your assignment questions: A short guide Academic skills short guide to assignments. Directive or task words: Tell you exactly what to do e.g., discuss, argue etc.. Subject matter: Specifically what you should be writing about.. Limiting words: Parts of the question that may narrow or alter the focus of your answer.. Example: To what extent can the novel White Teeth by Zadie Smith be read differently in the light of the 9/11 ... Assignment MCQ [Free PDF] Assignment Question 10. An assignment problem is solved to minimize the total processing time of four jobs (1, 2, 3 and 4) on four different machines such that each job is processed exactly by one machine and each machine processes exactly one job. The minimum total processing time is found to be 500 minutes. Answered: How should a graded journal assignment… A: Approach to solving the question: Comprehensively read the question and identify which answer best… Q: The problem with the Sophists according to Aristotle is that: O They break the law of… A: aristotle criticized the Sophists for breaking the law of non-contradiction, which states that… Understanding Trait Self-Esteem Assessment: Methods and Validity Assignment # 1 Instructions: Using the example of the Rosenberg Self-Esteem Scale (which is posted as a PDF), answer the questions that appear on the following page. Answers should be given directly underneath the grading rubric for each question, but do not necessarily have to "fit" in the space between each question (i.e., you can write a little bit more if you have to). Elon Musk's favorite job interview question is a secret method to catch You'll get the "X" if you fail this job interview question. Elon Musk, Space X founder and Tesla CEO revealed his favorite, time-tested job interview question that catches liars. During the ... Challenges in detecting ecological interactions using ... With increasing availability of ancient and modern environmental DNA technology, whole-community species occurrence and abundance data over time and space is becoming more available. Sedimentary ancient DNA data can be used to infer associations between species, which can generate hypotheses about biotic interactions, a key part of ecosystem function and biodiversity science. essay homework paper phd project resume review speech study thesis