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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.

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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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WHAT IS ASSIGNMENT PROBLEM

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 assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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Assignment Problem: Linear Programming

The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.

In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.

It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.

"A mathematician is a device for turning coffee into theorems." -Paul Erdos

Formulation of an assignment problem

Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:

The structure of an assignment problem is identical to that of a transportation problem.

To formulate the assignment problem in mathematical programming terms , we define the activity variables as

for i = 1, 2, ..., n and j = 1, 2, ..., n

In the above table, c ij is the cost of performing jth job by ith worker.

Generalized Form of an Assignment Problem

The optimization model is

Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn

subject to x i1 + x i2 +..........+ x in = 1          i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1          j = 1, 2,......., n

x ij = 0 or 1

In Σ Sigma notation

x ij = 0 or 1 for all i and j

An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.

Assumptions in Assignment Problem

• Number of jobs is equal to the number of machines or persons.
• Each man or machine is assigned only one job.
• Each man or machine is independently capable of handling any job to be done.
• Assigning criteria is clearly specified (minimizing cost or maximizing profit).

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Generalized Assignment Problem

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• O. Erhun Kundakcioglu 3 &
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Article Outline

Introduction

  Multiple-Resource Generalized Assignment Problem

  Multilevel Generalized Assignment Problem

  Dynamic Generalized Assignment Problem

  Bottleneck Generalized Assignment Problem

  Generalized Assignment Problem with Special Ordered Set

  Stochastic Generalized Assignment Problem

  Bi-Objective Generalized Assignment Problem

  Generalized Multi-Assignment Problem

  Exact Algorithms

  Heuristics

Conclusions

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Kundakcioglu, O.E., Alizamir, S. (2008). Generalized Assignment Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_200

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Basics of Assignment Structure and Format

Some students, particularly those in their freshman years, tend to overthink things and try to go for assignment structures that are unnecessarily complicated, thinking this will help them stand out from their peers and get better grades. It doesn’t have to be that hard.

This guide will give you an overview of basic assignment structure which you can use as a checklist for your assignments. This will help make sure that you haven’t missed any critical sections which are typically expected in assignment papers.

You will have to trust us when we say that your teachers will be really grateful that you’ve stuck to the standard format as it will make their grading process easier.

If you’re pressed for time, you can also head on over to our resources page to download some free assignment paper templates with generic outlines which you can tweak further to suit your needs. However, if you do happen to use any of these, then please ensure to follow our guide on checking document metadata details to avoid being flagged incorrectly for plagiarism.

• Table of Contents

Overview of basic assignment structure

A very common advice is that any written work, which includes assignments, should have an introduction, a body, and a conclusion . This is a form of oversimplification but should you give you general idea of what is expected. In reality, academic writing requires additional subheadings under in the body or main part of your text to convey your ideas in a structured way.

So, here’s a more specific overview of the main structural parts of an assignment.

• Introduction
• Literature Review
• Methodology
• Conclusions

We explain each of these in more details in the next sections

What are the essential parts in assignment structure?

The 4 parts of an academic writing work which should be considered essential are Introduction, Body, Conclusion and References . The last one should be obvious to any of our readers, but it’s surprising to see many students overlook the introduction and conclusion sections. Teachers often expect to see a short summary that sets the background and tone for the assignment, and they most definitely want to see what conclusions the student has reached by the end of their paper based on their study and research.

To make it simple, all you need do in the introduction is to give a brief overview of the topic which your paper is about, why this topic is relevant and important. In the conclusions section, you just need to summarize your research process, what you’ve learnt about the topic along the way and any final inferences.

These aren’t sections that you would have to do separate research for – if you’ve done your paper by yourself, you should be able to easily write a conclusion for it in no less than an hour (and we’re being very liberal with that estimation).

This is an easy to way to add in some additional words, which brings you closer to the required word count and reducing the words that you need to write for the other sections too, so why wouldn’t you want to do this, right?

As we have covered in our separate guide on how to manage word count of assignments , the introduction and conclusion sections are almost always included in the word count unless explicitly mentioned otherwise. You should be able to allocate about 200 to 300 words of the word count allowance to each of sections. This would cut down the amount of fresh content you need by about 400 to 600 words right off the bat!

Another critical section to be included in nearly all assignments would be a Table of Contents section. We have a full guide on how to easily make a good table of contents section which your teachers will be grateful to have when they are checking your work!

What are the circumstantial parts in assignment structure?

Moving on to the main part of your assignment, you could have a whole variety of headings and subheadings based on the type of paper that you are writing. Typically,

For thesis and dissertations, an Abstract section is almost always expected. Even if it hasn’t been specifically requested, we would highly recommend including this section for such long format papers because its purpose is to guide the readers with a ‘hook’ and make them more interested in reading your paper.

You can think of this section as a short summary of the main points from each of the broader headings in your paper. You don’t need to have more than 200 words for this section, and it shouldn’t be that hard to write as soon as you are done with your paper since the content that you’ve written should still be fresh in your mind.

The Rationale section is also expected in papers with longer word counts, especially those which are research oriented. In this section, you just need to explain the background of your chosen topic or research problem and why it is relevant and significant. You are expected to justify the need for your research on this topic. Some other research-oriented subsections include Research Aim & Objectives and Research Question , which you may potentially need, especially for long-form writing such as theses and dissertations.

Literature Review is a heading which can be considered as almost essential for most assignments since teachers want to see what external reading you have done on existing academic theory. The reason we have included this in the circumstantial section rather than the essential section above is because there are occasionally some assignments for which teachers explicitly mention not to define or explain academic theory and instead, they expect to see your understanding through direct application to the case.

This type of scenario typically arises when the word count for the assignment is not that high, so a full-blown literature review could be seen as ‘word-padding’ and have a negative impact on your overall grade for that assignment. We go over this issue of word-padding and how you can avoid it in a separate guide on managing word count as mentioned earlier.

A couple of sections that you can include in assignments where you are focusing on a particular industry or company are External Analysis and Internal Analysis respectively. As we cover in our guide on common academic words and what they mean, these are also known as Situational Analysis and Company Analysis , or Macro-analysis and Micro-analysis respectively. For company analysis, SWOT analysis one of the most common tools that are used, while another tool called TOWS is also occasionally used to combine internal and external analysis. Here’s a guide explaining the difference between SWOT and TOWS analysis .

Conceptual Framework is another circumstantial subsection which you may want to add if your research revolves around identifying independent and dependent variables.

For papers which require some research to be done as part of the assignment, you are going to need some sections like Methodology , Findings, Discussion and Analysis .

Within Methodology, you might have other subsections like Data Collection , Sampling Method etc.

What are some additional parts in assignment structure?

A section on Ethical Consent is often expected within the Research Methodology heading if your paper involves primary research gathered from respondents. We have a guide on ethical consent here and you can also check out our resources page to get a free sample ethical consent form which you could use in your papers (but be sure to also check out our guide on how to avoid getting flagged incorrectly for plagiarism so that you do not get wrongly flagged for plagiarism by using this sample form).

For research-oriented assignments, another typical expectation (although it isn’t always the case) is for a Research Timeline which illustrates the research process. This is often presented visually, and we have a separate guide on how to make good Gantt Charts easily using some of our free templates which you can find in our resources section.

Recommendations , and Limitations are also sometimes expected in some assignments, especially those that are long form, such as detailed research work.

If you have a lot of tables and figures in your work, which is usually the case for thesis and dissertations, you should also try to add a list of tables and figures (separate list for each).

Another section that you may wish to include in your assignments, especially if you have too many images or tables within your work, or if these are not to be included in the word count or in the body of the assignment, is the Appendix section. This is just a list of the additional content that is of direct relevance to your research adds to the quality and depth of your assignment. This section is typically not included in the word count allowance. Hence, it is often used to show that you have covered more ground in your research, but could not include it in main body due to word count limitation.

How to plan structure for essay assignments

Unlike standard academic reports, essays typically do not have subheadings but are instead expected to be structured in a logical way such that you transition from one idea to another by interlinking them.

However, we have come across some essay requirements in which the teachers have specifically asked for subheadings that indicate the central theme or idea which is being discussed in each section. If you are going to have subheadings in an essay though, then you should avoid numbering them as you would in other academic papers.

Taking assignment structures from the requirements files

Most teachers expect to see a standard structure in their assignments which helps them identify and assess the key learning outcomes of the module or the assignment. It is often the case that they will leave you some breadcrumbs by spelling out an assignment outline clearly within the requirements file, or in the presentations linked to the coursework, or perhaps during their lectures.

Sometimes, a structure would not have been defined clearly in the assignment requirements file, but there are ways to still get an almost explicit list of necessary headings such as by dissecting the marking guidelines which your tutors typically provide. We cover this in our guide on how to get better grades using a surefire method of drafting an outline which matches what the teachers are expecting.

Other teachers like to spice things up and give you free rein to structure your paper anyway you see it, but with this freedom of choice, you may find yourselves lost if you’re quite new to academic writing. In any of these cases, you can go over the upcoming list of standard sections and subsections that are present in academic papers to cross-check whether you have covered the relevant parts in your papers.

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Assignment structure

The structure of academic assignments often follows a standard outline.

However, depending on the topic of the assignment and the field of study, there may be some variation in the assignment structure. This page provides information about the typical parts of an academic assignment. The page may serve as inspiration on how to put your assignment together, but keep in mind that the structure should be adapted to fit your project, and not the other way around. 

Typical content elements 

The structure of your assignment depends, among other things, on whether it is a theoretical, empirical or product-oriented assignment. Read more on the page Types of assignments. Moreover, the structure should reflect that your assignment presents one overall argument supported by academic evidence. Read more about assignments as a single argument on the page Argumentation.  

Check your academic regulations

The content elements described below are typical parts of an academic assignment, but note that special requirements or recommendations may apply for the structure and content of the assignment you are writing. Therefore, you should always check your academic regulations, and possibly contact your supervisor or teacher at an early stage of the assignment process, so you can incorporate any specific requirements from the start. Be aware that the content elements described below may be called something else in your field of study. Use the terminology traditionally used on your degree programme. 

There is often a requirement for major university assignments to include an abstract or a brief summary, either at the beginning or at the end of the assignment. An abstract summarises the assignment’s: 

problem and objective 

methods 

analysis results 

conclusion 

perspectives 

An abstract gives the reader a quick insight into the assignment, so that they can assess whether it is relevant to read. 

Note : Not all assignments have to include an abstract. Check your academic regulations or ask your supervisor if you are in doubt. Be aware that the abstract may have to be written in another language than the rest of the assignment. 

Introduction

The introduction is where you present the framework of your assignment to your reader and provide an overview of what you want to achieve, and why. This includes a presentation of your topic and the problem you will be looking into, including the relevance of investigating it and how you will go about it.  

Edit the introduction continuously in the writing process and write it until the end to make sure that you do not promise more than the assignment provides. Ask yourself whether the conclusion responds to your problem statement, and whether the assignment contains all the aspects you promise in the introduction.

Problem statement/hypothesis

Regardless of whether you formulate it as a problem statement or a hypothesis, the problem addressed in your assignment should stand out clearly. For example, you can write it in italics, highlight it in bold or place it in a separate section with a heading. Read more about how to develop and work with a problem statement on the page Problem statement and hypothesis.  

The purpose of the assignment

The overall purpose of the assignment must be stated clearly in the introduction. Stating the purpose means explaining why your assignment is interesting to others and how it contributes to addressing the problem you are investigating. For example, your purpose could be: 

Research Overview/Literature Review

At university, you are expected to actively consider pre-existing knowledge about your topic and how it has previously been approached within your field of study. There are several ways to do this depending on the type of assignment and the subject you are studying.  

Sometimes you have to present existing research in a separate chapter or section where you discuss the latest research within the field and provide relevant literature reviews. And sometimes, a brief presentation of the most important research will be enough in either the introduction, theory section or elsewhere in the assignment.  

Check the academic regulations 

Check your academic regulations, or ask your supervisor or teacher about the requirements for including a research overview and pre-existing knowledge in your assignment. 

Click here to read more about the conventions for academic work  

Note : Not all written assignments have to include an actual research overview. Check your academic regulations or ask your supervisor if you are in doubt.  

Philosophy of science

Philosophy of science is a presentation of your approach to what knowledge is and how knowledge is produced. There are different scientific-theoretical schools of thought, with different views on what science is and ought to be. 

The schools of thought draw on different ontological understandings (i.e. understandings of how something exists) and different epistemological foundations (i.e. theories of knowledge and assumptions about the world). Examples of scientific-theoretical schools of thought are social constructivism, positivism, phenomenology and hermeneutics.  

Explain your scientific-theoretical approach 

Your scientific-theoretical approach must be based on philosophy of science literature and must be closely linked to your choices of methods and theories, which you may also elaborate on in this section. 

Read more about the use of pre-existing knowledge and independent conclusions on the page Academic standards.  

Methods and study design

The chapter on methodology and study design is a prerequisite for the validity of your investigation and analysis. Read more about this on the page about argumentation.  

Describe your study design 

The methodology section can vary depending on whether your assignment is theoretical, empirical or product-oriented. However, no matter what, it must include a description of how you conduct your study. This is also known as the study design. 

The study design refers to the overall framework for data collection and analysis. It should be based on the academic methods you have learned during class, and must be backed by theory of methods. 

Explain your choices and trade-offs 

Reflecting on and being conscious of the choices that you make is an important part of working academically. Therefore, in the methodology section, you should reflect on your conscious choices and the trade-offs you have had to make (for example due to external circumstances) and how this has affected your study design or your analysis. You can also explain why you have chosen a particular method if there were other obvious alternatives. 

The theory section is where you present and account for the theory used in the assignment. Make sure you take an application-oriented approach, i.e. account only for the theory that you actually use to answer your research questions further down in the analysis. Note that the purpose of the theory section is not to report everything you know about a particular field, but to support your study and your analysis as part of your argumentation . 

Different ways to integrate theory 

You can integrate the theory section in different ways. In some assignments, it makes most sense to have one separate theoretical chapter in which you explain all the theoretical concepts used in your assignment. In other assignments, it may make more sense to briefly present the theory in a separate section and then explain relevant theoretical concepts as they are applied in your analysis. Talk with your supervisor or your teacher about what would be most appropriate in your assignment. 

The analysis section of your assignment can take many different forms depending on whether your assignment is theoretical, empirical or product-oriented. The analysis is usually the most comprehensive part of the assignment because this is where you answer your research questions by presenting all your evidence for the overall claim of the assignment. 

Read more about argumentation.  

Guide your reader through the analysis 

Because the analysis is so comprehensive, it is a good idea to use meta-communication to guide the reader through the logic and progress of your assignment. For example, write sub-conclusions to sum up along the way. 

Read more about guiding your reader under Academic standards.  

Structure of the analysis 

In the analysis, the first thing you need to do is present the object, e.g. empirical data or artefacts, that you want to analyse and the tools you want to use for the analysis, e.g. your method, theory or concepts. Then you move on to the actual analysis, where you use the tools to examine the selected object of analysis. 

Note that it is difficult to write your analysis section before you have actually performed your analysis because you cannot see patterns, categories, etc. until you have the analysis material in front of you. 

Read more about the writing process  

In the video below, Master of Arts Rikke Gottfredsen explains what an analysis is (in Danish).

You can structure the analysis using the DAA structure: 

Description : Describe the sub-object you are about to analyse (e.g. a quotation or a table). 

Analysis : Analyse the sub-object using theories and concepts. 

Assessment : Assess what the analysis of the sub-object says about the overall object of analysis. 

The DAA structure can be repeated over and over again until all your sub-objects have been analysed. 

The discussion part of your assignment is where you criticise and defend your own study, both academically and methodologically. In other words, you have to consider the weaknesses of the assignment while demonstrating that, in spite of these, the assignment is still reliable. This will strengthen the overall argumentation of your assignment. 

Discuss your challenges

Encountering challenges during the writing process is quite common, and in some cases, they may serve as input for your discussion section. Note down challenges as they occur, including an explanation of why they occurred. In this way, you will have material for the discussion you are going to write later on. 

The conclusion summarises the results of your analysis and reiterates why the assignment is important. It must include clear and well-written answers to the research questions posed in your problem statement, or a confirmation or rejection of the hypothesis tested in your assignment. 

Depending on the purpose of your study, which was presented in the introduction, the conclusions may take different forms: 

Click here to read more about the characteristics of different purposes of investigation  

In addition to answering your research questions, or confirming or rejecting your hypothesis, the conclusion should also summarise the main points and results of the assignment. Moreover, it should include an assessment of your methodology and approach. 

The conclusion should never include new material, but should briefly summarise the main points of your study. It can be a good idea to write notes during your writing process that you can use for the conclusion. 

Is there a clear link between the introduction and the conclusion? 

Once you have finished writing your assignment, try to read the introduction and the conclusion in one go. Then assess whether the promises made in the introduction are being fulfilled in the conclusion, and whether the conclusion answers your research questions/hypothesis. 

Perspectives

In some university assignments, you are expected to end the assignment by discussing additional perspectives. The perspectives can be a separate section after the conclusion, they can form part of the conclusion, or they can be integrated into your discussion. Any perspectives should be based on what you have already written in the assignment. In other words, you should not integrate new theory or claims that require new evidence in your perspectives section. 

Click here to read about argumentation  

Check your academic regulations or talk to your supervisor or teacher if you are uncertain about whether your assignment should contain a perspectives section, and how it should be integrated into the assignment. 

Inspiration from assignments by other students

Get a list of thesis titles from your field of study, and draw inspiration from other students’ assignments. 

Avoid cheating in your assignment

It is important to follow the  rules and guidelines on exam cheating and plagiarism . AU Library guides you on how, so you can easily avoid it.