## 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:

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## Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

[a1] | D.B. Yudin, E.G. Gol'shtein, "Linear programming" , Israel Program Sci. Transl. (1965) (In Russian) |

[a2] | R. Frisch, "La résolution des problèmes de programme linéaire par la méthode du potentiel logarithmique" , (1956) pp. 20–23 |

[a3] | K. Murtz, "Linear and combinatorial programming" , Wiley (1976) |

[a4] | M. Grötschel, L. Lovász, A. Schrijver, "Geometric algorithms and combinatorial optimization" , Springer (1987) |

[a5] | C.H. Papadimitriou, K. Steiglitz, "Combinatorial optimization" , Prentice-Hall (1982) |

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## Operations Research - Definition and formulation of Assignment Problem | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research, definition and formulation of assignment problem.

Definition and formulation

Consider the problem of assigning n jobs to n machines (one job to one machine). Let C ij be the cost of assigning i th job to the j th machine and x ij represents the assignment of i th job to the j th machine.

x ij is missing in any cell means that no assignment is made between the pair of job and machine.( i.e ) x ij = 0.

x ij presents in any cell means that an assignment is made their.In such cases x ij = 1

The assignment model can be written in LPP as follows

Subject to the constrains

The optimum assignment schedule remains unaltered if we add or subtract a constant from all the elements of the row or column of the assignment cost matrix.

If for an assignment problem all C ij > 0 then an assignment schedule (x ij ) which satisfies ∑ C ij x ij = 0 must be optimal.

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Statistics By Jim

Making statistics intuitive

## Random Assignment in Experiments

By Jim Frost 4 Comments

Random assignment uses chance to assign subjects to the control and treatment groups in an experiment. This process helps ensure that the groups are equivalent at the beginning of the study, which makes it safer to assume the treatments caused any differences between groups that the experimenters observe at the end of the study.

Huh? That might be a big surprise! At this point, you might be wondering about all of those studies that use statistics to assess the effects of different treatments. There’s a critical separation between significance and causality:

- Statistical procedures determine whether an effect is significant.
- Experimental designs determine how confidently you can assume that a treatment causes the effect.

In this post, learn how using random assignment in experiments can help you identify causal relationships.

## Correlation, Causation, and Confounding Variables

Random assignment helps you separate causation from correlation and rule out confounding variables. As a critical component of the scientific method , experiments typically set up contrasts between a control group and one or more treatment groups. The idea is to determine whether the effect, which is the difference between a treatment group and the control group, is statistically significant. If the effect is significant, group assignment correlates with different outcomes.

However, as you have no doubt heard, correlation does not necessarily imply causation. In other words, the experimental groups can have different mean outcomes, but the treatment might not be causing those differences even though the differences are statistically significant.

The difficulty in definitively stating that a treatment caused the difference is due to potential confounding variables or confounders. Confounders are alternative explanations for differences between the experimental groups. Confounding variables correlate with both the experimental groups and the outcome variable. In this situation, confounding variables can be the actual cause for the outcome differences rather than the treatments themselves. As you’ll see, if an experiment does not account for confounding variables, they can bias the results and make them untrustworthy.

Related posts : Understanding Correlation in Statistics , Causation versus Correlation , and Hill’s Criteria for Causation .

## Example of Confounding in an Experiment

- Control group: Does not consume vitamin supplements
- Treatment group: Regularly consumes vitamin supplements.

Imagine we measure a specific health outcome. After the experiment is complete, we perform a 2-sample t-test to determine whether the mean outcomes for these two groups are different. Assume the test results indicate that the mean health outcome in the treatment group is significantly better than the control group.

Why can’t we assume that the vitamins improved the health outcomes? After all, only the treatment group took the vitamins.

Related post : Confounding Variables in Regression Analysis

## Alternative Explanations for Differences in Outcomes

The answer to that question depends on how we assigned the subjects to the experimental groups. If we let the subjects decide which group to join based on their existing vitamin habits, it opens the door to confounding variables. It’s reasonable to assume that people who take vitamins regularly also tend to have other healthy habits. These habits are confounders because they correlate with both vitamin consumption (experimental group) and the health outcome measure.

Random assignment prevents this self sorting of participants and reduces the likelihood that the groups start with systematic differences.

In fact, studies have found that supplement users are more physically active, have healthier diets, have lower blood pressure, and so on compared to those who don’t take supplements. If subjects who already take vitamins regularly join the treatment group voluntarily, they bring these healthy habits disproportionately to the treatment group. Consequently, these habits will be much more prevalent in the treatment group than the control group.

The healthy habits are the confounding variables—the potential alternative explanations for the difference in our study’s health outcome. It’s entirely possible that these systematic differences between groups at the start of the study might cause the difference in the health outcome at the end of the study—and not the vitamin consumption itself!

If our experiment doesn’t account for these confounding variables, we can’t trust the results. While we obtained statistically significant results with the 2-sample t-test for health outcomes, we don’t know for sure whether the vitamins, the systematic difference in habits, or some combination of the two caused the improvements.

Learn why many randomized clinical experiments use a placebo to control for the Placebo Effect .

## Experiments Must Account for Confounding Variables

Your experimental design must account for confounding variables to avoid their problems. Scientific studies commonly use the following methods to handle confounders:

- Use control variables to keep them constant throughout an experiment.
- Statistically control for them in an observational study.
- Use random assignment to reduce the likelihood that systematic differences exist between experimental groups when the study begins.

Let’s take a look at how random assignment works in an experimental design.

## Random Assignment Can Reduce the Impact of Confounding Variables

Note that random assignment is different than random sampling. Random sampling is a process for obtaining a sample that accurately represents a population .

Random assignment uses a chance process to assign subjects to experimental groups. Using random assignment requires that the experimenters can control the group assignment for all study subjects. For our study, we must be able to assign our participants to either the control group or the supplement group. Clearly, if we don’t have the ability to assign subjects to the groups, we can’t use random assignment!

Additionally, the process must have an equal probability of assigning a subject to any of the groups. For example, in our vitamin supplement study, we can use a coin toss to assign each subject to either the control group or supplement group. For more complex experimental designs, we can use a random number generator or even draw names out of a hat.

## Random Assignment Distributes Confounders Equally

The random assignment process distributes confounding properties amongst your experimental groups equally. In other words, randomness helps eliminate systematic differences between groups. For our study, flipping the coin tends to equalize the distribution of subjects with healthier habits between the control and treatment group. Consequently, these two groups should start roughly equal for all confounding variables, including healthy habits!

Random assignment is a simple, elegant solution to a complex problem. For any given study area, there can be a long list of confounding variables that you could worry about. However, using random assignment, you don’t need to know what they are, how to detect them, or even measure them. Instead, use random assignment to equalize them across your experimental groups so they’re not a problem.

Because random assignment helps ensure that the groups are comparable when the experiment begins, you can be more confident that the treatments caused the post-study differences. Random assignment helps increase the internal validity of your study.

## Comparing the Vitamin Study With and Without Random Assignment

Let’s compare two scenarios involving our hypothetical vitamin study. We’ll assume that the study obtains statistically significant results in both cases.

Scenario 1: We don’t use random assignment and, unbeknownst to us, subjects with healthier habits disproportionately end up in the supplement treatment group. The experimental groups differ by both healthy habits and vitamin consumption. Consequently, we can’t determine whether it was the habits or vitamins that improved the outcomes.

Scenario 2: We use random assignment and, consequently, the treatment and control groups start with roughly equal levels of healthy habits. The intentional introduction of vitamin supplements in the treatment group is the primary difference between the groups. Consequently, we can more confidently assert that the supplements caused an improvement in health outcomes.

For both scenarios, the statistical results could be identical. However, the methodology behind the second scenario makes a stronger case for a causal relationship between vitamin supplement consumption and health outcomes.

How important is it to use the correct methodology? Well, if the relationship between vitamins and health outcomes is not causal, then consuming vitamins won’t cause your health outcomes to improve regardless of what the study indicates. Instead, it’s probably all the other healthy habits!

Learn more about Randomized Controlled Trials (RCTs) that are the gold standard for identifying causal relationships because they use random assignment.

## Drawbacks of Random Assignment

Random assignment helps reduce the chances of systematic differences between the groups at the start of an experiment and, thereby, mitigates the threats of confounding variables and alternative explanations. However, the process does not always equalize all of the confounding variables. Its random nature tends to eliminate systematic differences, but it doesn’t always succeed.

Sometimes random assignment is impossible because the experimenters cannot control the treatment or independent variable. For example, if you want to determine how individuals with and without depression perform on a test, you cannot randomly assign subjects to these groups. The same difficulty occurs when you’re studying differences between genders.

In other cases, there might be ethical issues. For example, in a randomized experiment, the researchers would want to withhold treatment for the control group. However, if the treatments are vaccinations, it might be unethical to withhold the vaccinations.

Other times, random assignment might be possible, but it is very challenging. For example, with vitamin consumption, it’s generally thought that if vitamin supplements cause health improvements, it’s only after very long-term use. It’s hard to enforce random assignment with a strict regimen for usage in one group and non-usage in the other group over the long-run. Or imagine a study about smoking. The researchers would find it difficult to assign subjects to the smoking and non-smoking groups randomly!

Fortunately, if you can’t use random assignment to help reduce the problem of confounding variables, there are different methods available. The other primary approach is to perform an observational study and incorporate the confounders into the statistical model itself. For more information, read my post Observational Studies Explained .

## Read About Real Experiments that Used Random Assignment

I’ve written several blog posts about studies that have used random assignment to make causal inferences. Read studies about the following:

- Flu Vaccinations
- COVID-19 Vaccinations

Sullivan L. Random assignment versus random selection . SAGE Glossary of the Social and Behavioral Sciences, SAGE Publications, Inc.; 2009.

## Share this:

## Reader Interactions

November 13, 2019 at 4:59 am

Hi Jim, I have a question of randomly assigning participants to one of two conditions when it is an ongoing study and you are not sure of how many participants there will be. I am using this random assignment tool for factorial experiments. http://methodologymedia.psu.edu/most/rannumgenerator It asks you for the total number of participants but at this point, I am not sure how many there will be. Thanks for any advice you can give me, Floyd

May 28, 2019 at 11:34 am

Jim, can you comment on the validity of using the following approach when we can’t use random assignments. I’m in education, we have an ACT prep course that we offer. We can’t force students to take it and we can’t keep them from taking it either. But we want to know if it’s working. Let’s say that by senior year all students who are going to take the ACT have taken it. Let’s also say that I’m only including students who have taking it twice (so I can show growth between first and second time taking it). What I’ve done to address confounders is to go back to say 8th or 9th grade (prior to anyone taking the ACT or the ACT prep course) and run an analysis showing the two groups are not significantly different to start with. Is this valid? If the ACT prep students were higher achievers in 8th or 9th grade, I could not assume my prep course is effecting greater growth, but if they were not significantly different in 8th or 9th grade, I can assume the significant difference in ACT growth (from first to second testing) is due to the prep course. Yes or no?

May 26, 2019 at 5:37 pm

Nice post! I think the key to understanding scientific research is to understand randomization. And most people don’t get it.

May 27, 2019 at 9:48 pm

Thank you, Anoop!

I think randomness in an experiment is a funny thing. The issue of confounding factors is a serious problem. You might not even know what they are! But, use random assignment and, voila, the problem usually goes away! If you can’t use random assignment, suddenly you have a whole host of issues to worry about, which I’ll be writing about in more detail in my upcoming post about observational experiments!

## Comments and Questions Cancel reply

## 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:

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 5 | 7 | 6 |

Emp 2 | 6 | 4 | 5 |

Emp 3 | 8 | 5 | 3 |

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:

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 0 | 2 | 1 |

Emp 2 | 2 | 0 | 1 |

Emp 3 | 5 | 2 | 0 |

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

Task 1 | Task 2 | Task 3 | |
---|---|---|---|

Emp 1 | 0 | 2 | 1 |

Emp 2 | 2 | 0 | 1 |

Emp 3 | 5 | 2 | 0 |

0 | 0 | 0 |

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

## A Comparative Analysis of Assignment Problem

- Conference paper
- First Online: 06 June 2023
- Cite this conference paper

- Shahriar Tanvir Alam ORCID: orcid.org/0000-0002-0567-3381 5 ,
- Eshfar Sagor 5 ,
- Tanjeel Ahmed 5 ,
- Tabassum Haque 5 ,
- Md Shoaib Mahmud 5 ,
- Salman Ibrahim 5 ,
- Ononya Shahjahan 5 &
- Mubtasim Rubaet 5

Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))

Included in the following conference series:

- International Conference on Big Data Innovation for Sustainable Cognitive Computing

144 Accesses

The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.

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Shahriar Tanvir Alam, Eshfar Sagor, Tanjeel Ahmed, Tabassum Haque, Md Shoaib Mahmud, Salman Ibrahim, Ononya Shahjahan & Mubtasim Rubaet

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Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11

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Methodology

- Random Assignment in Experiments | Introduction & Examples

## Random Assignment in Experiments | Introduction & Examples

Published on March 8, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In experimental research, random assignment is a way of placing participants from your sample into different treatment groups using randomization.

With simple random assignment, every member of the sample has a known or equal chance of being placed in a control group or an experimental group. Studies that use simple random assignment are also called completely randomized designs .

Random assignment is a key part of experimental design . It helps you ensure that all groups are comparable at the start of a study: any differences between them are due to random factors, not research biases like sampling bias or selection bias .

## Table of contents

Why does random assignment matter, random sampling vs random assignment, how do you use random assignment, when is random assignment not used, other interesting articles, frequently asked questions about random assignment.

Random assignment is an important part of control in experimental research, because it helps strengthen the internal validity of an experiment and avoid biases.

In experiments, researchers manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. To do so, they often use different levels of an independent variable for different groups of participants.

This is called a between-groups or independent measures design.

You use three groups of participants that are each given a different level of the independent variable:

- a control group that’s given a placebo (no dosage, to control for a placebo effect ),
- an experimental group that’s given a low dosage,
- a second experimental group that’s given a high dosage.

Random assignment to helps you make sure that the treatment groups don’t differ in systematic ways at the start of the experiment, as this can seriously affect (and even invalidate) your work.

If you don’t use random assignment, you may not be able to rule out alternative explanations for your results.

- participants recruited from cafes are placed in the control group ,
- participants recruited from local community centers are placed in the low dosage experimental group,
- participants recruited from gyms are placed in the high dosage group.

With this type of assignment, it’s hard to tell whether the participant characteristics are the same across all groups at the start of the study. Gym-users may tend to engage in more healthy behaviors than people who frequent cafes or community centers, and this would introduce a healthy user bias in your study.

Although random assignment helps even out baseline differences between groups, it doesn’t always make them completely equivalent. There may still be extraneous variables that differ between groups, and there will always be some group differences that arise from chance.

Most of the time, the random variation between groups is low, and, therefore, it’s acceptable for further analysis. This is especially true when you have a large sample. In general, you should always use random assignment in experiments when it is ethically possible and makes sense for your study topic.

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Random sampling and random assignment are both important concepts in research, but it’s important to understand the difference between them.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups.

While random sampling is used in many types of studies, random assignment is only used in between-subjects experimental designs.

Some studies use both random sampling and random assignment, while others use only one or the other.

Random sampling enhances the external validity or generalizability of your results, because it helps ensure that your sample is unbiased and representative of the whole population. This allows you to make stronger statistical inferences .

You use a simple random sample to collect data. Because you have access to the whole population (all employees), you can assign all 8000 employees a number and use a random number generator to select 300 employees. These 300 employees are your full sample.

Random assignment enhances the internal validity of the study, because it ensures that there are no systematic differences between the participants in each group. This helps you conclude that the outcomes can be attributed to the independent variable .

- a control group that receives no intervention.
- an experimental group that has a remote team-building intervention every week for a month.

You use random assignment to place participants into the control or experimental group. To do so, you take your list of participants and assign each participant a number. Again, you use a random number generator to place each participant in one of the two groups.

To use simple random assignment, you start by giving every member of the sample a unique number. Then, you can use computer programs or manual methods to randomly assign each participant to a group.

- Random number generator: Use a computer program to generate random numbers from the list for each group.
- Lottery method: Place all numbers individually in a hat or a bucket, and draw numbers at random for each group.
- Flip a coin: When you only have two groups, for each number on the list, flip a coin to decide if they’ll be in the control or the experimental group.
- Use a dice: When you have three groups, for each number on the list, roll a dice to decide which of the groups they will be in. For example, assume that rolling 1 or 2 lands them in a control group; 3 or 4 in an experimental group; and 5 or 6 in a second control or experimental group.

This type of random assignment is the most powerful method of placing participants in conditions, because each individual has an equal chance of being placed in any one of your treatment groups.

## Random assignment in block designs

In more complicated experimental designs, random assignment is only used after participants are grouped into blocks based on some characteristic (e.g., test score or demographic variable). These groupings mean that you need a larger sample to achieve high statistical power .

For example, a randomized block design involves placing participants into blocks based on a shared characteristic (e.g., college students versus graduates), and then using random assignment within each block to assign participants to every treatment condition. This helps you assess whether the characteristic affects the outcomes of your treatment.

In an experimental matched design , you use blocking and then match up individual participants from each block based on specific characteristics. Within each matched pair or group, you randomly assign each participant to one of the conditions in the experiment and compare their outcomes.

Sometimes, it’s not relevant or ethical to use simple random assignment, so groups are assigned in a different way.

## When comparing different groups

Sometimes, differences between participants are the main focus of a study, for example, when comparing men and women or people with and without health conditions. Participants are not randomly assigned to different groups, but instead assigned based on their characteristics.

In this type of study, the characteristic of interest (e.g., gender) is an independent variable, and the groups differ based on the different levels (e.g., men, women, etc.). All participants are tested the same way, and then their group-level outcomes are compared.

## When it’s not ethically permissible

When studying unhealthy or dangerous behaviors, it’s not possible to use random assignment. For example, if you’re studying heavy drinkers and social drinkers, it’s unethical to randomly assign participants to one of the two groups and ask them to drink large amounts of alcohol for your experiment.

When you can’t assign participants to groups, you can also conduct a quasi-experimental study . In a quasi-experiment, you study the outcomes of pre-existing groups who receive treatments that you may not have any control over (e.g., heavy drinkers and social drinkers). These groups aren’t randomly assigned, but may be considered comparable when some other variables (e.g., age or socioeconomic status) are controlled for.

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If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

- Student’s t -distribution
- Normal distribution
- Null and Alternative Hypotheses
- Chi square tests
- Confidence interval
- Quartiles & Quantiles
- Cluster sampling
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- Peer review
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Research bias

- Implicit bias
- Cognitive bias
- Placebo effect
- Hawthorne effect
- Hindsight bias
- Affect heuristic
- Social desirability bias

In experimental research, random assignment is a way of placing participants from your sample into different groups using randomization. With this method, every member of the sample has a known or equal chance of being placed in a control group or an experimental group.

Random selection, or random sampling , is a way of selecting members of a population for your study’s sample.

In contrast, random assignment is a way of sorting the sample into control and experimental groups.

Random sampling enhances the external validity or generalizability of your results, while random assignment improves the internal validity of your study.

Random assignment is used in experiments with a between-groups or independent measures design. In this research design, there’s usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable.

In general, you should always use random assignment in this type of experimental design when it is ethically possible and makes sense for your study topic.

To implement random assignment , assign a unique number to every member of your study’s sample .

Then, you can use a random number generator or a lottery method to randomly assign each number to a control or experimental group. You can also do so manually, by flipping a coin or rolling a dice to randomly assign participants to groups.

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## Assignment Problem

5.1 introduction.

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.

## 5.2 GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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## COMMENTS

The formal definition of the assignment problem (or linear assignment problem) is . Given two sets, A and T, together with a weight function C : A × T → R.Find a bijection f : A → T such that the cost function: (, ())is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as: , The problem is "linear" because the cost ...

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 ...

Definition of Assignment Problem. The statement of the assignment problem is as follows: There are n men and n jobs, with a cost c, for assigning man i to job j. It is required to assign all men to jobs such that one and only one man is assigned to each job and the total cost of the assignments is minimal.

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 assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $. If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem). In the assignment problem, for such ...

The assignment problem represents a special case of linear programming problem used for allocating resources (mostly workforce) in an optimal way; it is a highly useful tool for operation and project managers for optimizing costs. The lpSolve R package allows us to solve LP assignment problems with just very few lines of code.

Assignment problems deal with the question of how to assign other items (machines, tasks). There are different ways in mathematics to describe an assignment: we can view an assignment as a bijective mapping φ between two finite sets . We can write a permutation φ as 12…nφ (1)φ (2)…φ (n), which means that 1 is mapped to φ (1), 2 is ...

The formal definition of the quadratic assignment problem is as follows: Given two sets, P ("facilities") and L ("locations"), of equal size, together with a weight function w : P × P → R and a distance function d : L × L → R.Find the bijection f : P → L ("assignment") such that the cost function:, (,) ((), ()) is minimized. Usually weight and distance functions are viewed as square ...

The problem is a special form of the transportation problem and, as such, has an optimal solution in which each variable is either zero or one. The problem can be solved by the simplex method, but special assignment problem algorithms tend to be computationally more efficient.

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to ...

Definition and formulation. Consider the problem of assigning n jobs to n machines (one job to one machine). Let Cij be the cost of assigning ith job to the jth machine and xij represents the assignment of ith job to the jth machine. xij is missing in any cell means that no assignment is made between the pair of job and machine. (i.e) xij = 0.

The Multicommodity Multilevel Bottleneck Assignment Problem. The Multilevel Bottleneck Assignment Problem is defined on a weighted graph of L levels and consists in finding L−1L−1 complete matchings between contiguous levels, such that the heaviest path formed by the arcs in the matchings has a minimum weight.

Statistics is a study of data: describing properties of data (descriptive statistics) and drawing conclusions about a population based on information in a sample (inferential statistics). The distinction between a population together with its parameters and a sample together with its statistics is a fundamental concept in inferential statistics ...

Random assignment: Can determine causal relationship in population. This design is relatively rare in the real world. Can determine causal relationship in that sample only. This design is where most experiments would fit. No random assignment: Can detect relationships in population, but cannot determine causality.

Correlation, Causation, and Confounding Variables. Random assignment helps you separate causation from correlation and rule out confounding variables. As a critical component of the scientific method, experiments typically set up contrasts between a control group and one or more treatment groups. The idea is to determine whether the effect, which is the difference between a treatment group and ...

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.

Unit 7: Probability. Unit mastery: 0%. Basic theoretical probability: Probability Probability using sample spaces: Probability Basic set operations: Probability Experimental probability: Probability. Randomness, probability, and simulation: Probability Addition rule: Probability Multiplication rule for independent events: Probability ...

Statistics is a branch of mathematics used to summarize, analyze, and interpret a group of numbers or observations. We begin by introducing two general types of statistics: •• Descriptive statistics: statistics that summarize observations. •• Inferential statistics: statistics used to interpret the meaning of descriptive statistics.

assignment problem occurs frequently in practice and is a basic problem in network ﬂow theory since it can be reduced to a number of other problems, including the shortest path, weighted matching, transportation, and minimal cost ﬂow [4]. Furthermore, a World Bank-funded review estimated that in the "Greater Cai-ro ...

Table of contents. Step 1: Write your hypotheses and plan your research design. Step 2: Collect data from a sample. Step 3: Summarize your data with descriptive statistics. Step 4: Test hypotheses or make estimates with inferential statistics.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups. While random sampling is used in many types of studies, random assignment is only used ...

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the ...

In addition to being a useful tool for assessing statistical thinking, the four-step structure may have helped improve students' statistical thinking, their writing skill, or both, as the scores on the writing assignments increased on average from the beginning of the semester until the end. In. Figures 7. and. 8.