Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

For example:

Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

The cost of performing each task by each resource is also known (shown in cells of matrix)

Fig 1-assigment model intro

  • In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model.

→ Assignment Model is a special application of Linear Programming (LP).

→ The mathematical formulation for Assignment Model is given below:

→ Let, C i j \text {C}_{ij} C ij ​ denotes the cost of resources 'i' to the task 'j' ; such that

assignment model examples

→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

→ Subjected to constraint;

(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:

(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;

(iii) x i j x_{ij} x ij ​ is '0' or '1'.

Types of Assignment Problem:

(i) balanced assignment problem.

  • It consist of a suqare matrix (n x n).
  • Number of rows = Number of columns

(ii) Unbalanced Assignment Problem

  • It consist of a Non-square matrix.
  • Number of rows ≠ \not=  = Number of columns

Methods to solve Assignment Model:

(i) integer programming method:.

In assignment problem, either allocation is done to the cell or not.

So this can be formulated using 0 or 1 integer.

While using this method, we will have n x n decision varables, and n+n equalities.

So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

But, here in assignment problems, the matrix is a square matrix (m=n).

So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.

The assignments which gives minimum cost is selected as optimal solution.

But, such trail and error becomes very difficult and lengthy.

If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.

(iv) Hungarian Method:

It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

It is also know as Reduced matrix method or Flood's technique.

There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n). (2) Problem should be of minimization type.

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Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.

In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .

The costs of assigning workers to tasks are shown in the following table.

The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.

MIP solution

The following sections describe how to solve the problem using the MPSolver wrapper .

Import the libraries

The following code imports the required libraries.

Create the data

The following code creates the data for the problem.

The costs array corresponds to the table of costs for assigning workers to tasks, shown above.

Declare the MIP solver

The following code declares the MIP solver.

Create the variables

The following code creates binary integer variables for the problem.

Create the constraints

Create the objective function.

The following code creates the objective function for the problem.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

Invoke the solver

The following code invokes the solver.

Print the solution

The following code prints the solution to the problem.

Here is the output of the program.

Complete programs

Here are the complete programs for the MIP solution.

CP SAT solution

The following sections describe how to solve the problem using the CP-SAT solver.

Declare the model

The following code declares the CP-SAT model.

The following code sets up the data for the problem.

The following code creates the constraints for the problem.

Here are the complete programs for the CP-SAT solution.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.

The Writing Center • University of North Carolina at Chapel Hill

Understanding Assignments

What this handout is about.

The first step in any successful college writing venture is reading the assignment. While this sounds like a simple task, it can be a tough one. This handout will help you unravel your assignment and begin to craft an effective response. Much of the following advice will involve translating typical assignment terms and practices into meaningful clues to the type of writing your instructor expects. See our short video for more tips.

Basic beginnings

Regardless of the assignment, department, or instructor, adopting these two habits will serve you well :

  • Read the assignment carefully as soon as you receive it. Do not put this task off—reading the assignment at the beginning will save you time, stress, and problems later. An assignment can look pretty straightforward at first, particularly if the instructor has provided lots of information. That does not mean it will not take time and effort to complete; you may even have to learn a new skill to complete the assignment.
  • Ask the instructor about anything you do not understand. Do not hesitate to approach your instructor. Instructors would prefer to set you straight before you hand the paper in. That’s also when you will find their feedback most useful.

Assignment formats

Many assignments follow a basic format. Assignments often begin with an overview of the topic, include a central verb or verbs that describe the task, and offer some additional suggestions, questions, or prompts to get you started.

An Overview of Some Kind

The instructor might set the stage with some general discussion of the subject of the assignment, introduce the topic, or remind you of something pertinent that you have discussed in class. For example:

“Throughout history, gerbils have played a key role in politics,” or “In the last few weeks of class, we have focused on the evening wear of the housefly …”

The Task of the Assignment

Pay attention; this part tells you what to do when you write the paper. Look for the key verb or verbs in the sentence. Words like analyze, summarize, or compare direct you to think about your topic in a certain way. Also pay attention to words such as how, what, when, where, and why; these words guide your attention toward specific information. (See the section in this handout titled “Key Terms” for more information.)

“Analyze the effect that gerbils had on the Russian Revolution”, or “Suggest an interpretation of housefly undergarments that differs from Darwin’s.”

Additional Material to Think about

Here you will find some questions to use as springboards as you begin to think about the topic. Instructors usually include these questions as suggestions rather than requirements. Do not feel compelled to answer every question unless the instructor asks you to do so. Pay attention to the order of the questions. Sometimes they suggest the thinking process your instructor imagines you will need to follow to begin thinking about the topic.

“You may wish to consider the differing views held by Communist gerbils vs. Monarchist gerbils, or Can there be such a thing as ‘the housefly garment industry’ or is it just a home-based craft?”

These are the instructor’s comments about writing expectations:

“Be concise”, “Write effectively”, or “Argue furiously.”

Technical Details

These instructions usually indicate format rules or guidelines.

“Your paper must be typed in Palatino font on gray paper and must not exceed 600 pages. It is due on the anniversary of Mao Tse-tung’s death.”

The assignment’s parts may not appear in exactly this order, and each part may be very long or really short. Nonetheless, being aware of this standard pattern can help you understand what your instructor wants you to do.

Interpreting the assignment

Ask yourself a few basic questions as you read and jot down the answers on the assignment sheet:

Why did your instructor ask you to do this particular task?

Who is your audience.

  • What kind of evidence do you need to support your ideas?

What kind of writing style is acceptable?

  • What are the absolute rules of the paper?

Try to look at the question from the point of view of the instructor. Recognize that your instructor has a reason for giving you this assignment and for giving it to you at a particular point in the semester. In every assignment, the instructor has a challenge for you. This challenge could be anything from demonstrating an ability to think clearly to demonstrating an ability to use the library. See the assignment not as a vague suggestion of what to do but as an opportunity to show that you can handle the course material as directed. Paper assignments give you more than a topic to discuss—they ask you to do something with the topic. Keep reminding yourself of that. Be careful to avoid the other extreme as well: do not read more into the assignment than what is there.

Of course, your instructor has given you an assignment so that he or she will be able to assess your understanding of the course material and give you an appropriate grade. But there is more to it than that. Your instructor has tried to design a learning experience of some kind. Your instructor wants you to think about something in a particular way for a particular reason. If you read the course description at the beginning of your syllabus, review the assigned readings, and consider the assignment itself, you may begin to see the plan, purpose, or approach to the subject matter that your instructor has created for you. If you still aren’t sure of the assignment’s goals, try asking the instructor. For help with this, see our handout on getting feedback .

Given your instructor’s efforts, it helps to answer the question: What is my purpose in completing this assignment? Is it to gather research from a variety of outside sources and present a coherent picture? Is it to take material I have been learning in class and apply it to a new situation? Is it to prove a point one way or another? Key words from the assignment can help you figure this out. Look for key terms in the form of active verbs that tell you what to do.

Key Terms: Finding Those Active Verbs

Here are some common key words and definitions to help you think about assignment terms:

Information words Ask you to demonstrate what you know about the subject, such as who, what, when, where, how, and why.

  • define —give the subject’s meaning (according to someone or something). Sometimes you have to give more than one view on the subject’s meaning
  • describe —provide details about the subject by answering question words (such as who, what, when, where, how, and why); you might also give details related to the five senses (what you see, hear, feel, taste, and smell)
  • explain —give reasons why or examples of how something happened
  • illustrate —give descriptive examples of the subject and show how each is connected with the subject
  • summarize —briefly list the important ideas you learned about the subject
  • trace —outline how something has changed or developed from an earlier time to its current form
  • research —gather material from outside sources about the subject, often with the implication or requirement that you will analyze what you have found

Relation words Ask you to demonstrate how things are connected.

  • compare —show how two or more things are similar (and, sometimes, different)
  • contrast —show how two or more things are dissimilar
  • apply—use details that you’ve been given to demonstrate how an idea, theory, or concept works in a particular situation
  • cause —show how one event or series of events made something else happen
  • relate —show or describe the connections between things

Interpretation words Ask you to defend ideas of your own about the subject. Do not see these words as requesting opinion alone (unless the assignment specifically says so), but as requiring opinion that is supported by concrete evidence. Remember examples, principles, definitions, or concepts from class or research and use them in your interpretation.

  • assess —summarize your opinion of the subject and measure it against something
  • prove, justify —give reasons or examples to demonstrate how or why something is the truth
  • evaluate, respond —state your opinion of the subject as good, bad, or some combination of the two, with examples and reasons
  • support —give reasons or evidence for something you believe (be sure to state clearly what it is that you believe)
  • synthesize —put two or more things together that have not been put together in class or in your readings before; do not just summarize one and then the other and say that they are similar or different—you must provide a reason for putting them together that runs all the way through the paper
  • analyze —determine how individual parts create or relate to the whole, figure out how something works, what it might mean, or why it is important
  • argue —take a side and defend it with evidence against the other side

More Clues to Your Purpose As you read the assignment, think about what the teacher does in class:

  • What kinds of textbooks or coursepack did your instructor choose for the course—ones that provide background information, explain theories or perspectives, or argue a point of view?
  • In lecture, does your instructor ask your opinion, try to prove her point of view, or use keywords that show up again in the assignment?
  • What kinds of assignments are typical in this discipline? Social science classes often expect more research. Humanities classes thrive on interpretation and analysis.
  • How do the assignments, readings, and lectures work together in the course? Instructors spend time designing courses, sometimes even arguing with their peers about the most effective course materials. Figuring out the overall design to the course will help you understand what each assignment is meant to achieve.

Now, what about your reader? Most undergraduates think of their audience as the instructor. True, your instructor is a good person to keep in mind as you write. But for the purposes of a good paper, think of your audience as someone like your roommate: smart enough to understand a clear, logical argument, but not someone who already knows exactly what is going on in your particular paper. Remember, even if the instructor knows everything there is to know about your paper topic, he or she still has to read your paper and assess your understanding. In other words, teach the material to your reader.

Aiming a paper at your audience happens in two ways: you make decisions about the tone and the level of information you want to convey.

  • Tone means the “voice” of your paper. Should you be chatty, formal, or objective? Usually you will find some happy medium—you do not want to alienate your reader by sounding condescending or superior, but you do not want to, um, like, totally wig on the man, you know? Eschew ostentatious erudition: some students think the way to sound academic is to use big words. Be careful—you can sound ridiculous, especially if you use the wrong big words.
  • The level of information you use depends on who you think your audience is. If you imagine your audience as your instructor and she already knows everything you have to say, you may find yourself leaving out key information that can cause your argument to be unconvincing and illogical. But you do not have to explain every single word or issue. If you are telling your roommate what happened on your favorite science fiction TV show last night, you do not say, “First a dark-haired white man of average height, wearing a suit and carrying a flashlight, walked into the room. Then a purple alien with fifteen arms and at least three eyes turned around. Then the man smiled slightly. In the background, you could hear a clock ticking. The room was fairly dark and had at least two windows that I saw.” You also do not say, “This guy found some aliens. The end.” Find some balance of useful details that support your main point.

You’ll find a much more detailed discussion of these concepts in our handout on audience .

The Grim Truth

With a few exceptions (including some lab and ethnography reports), you are probably being asked to make an argument. You must convince your audience. It is easy to forget this aim when you are researching and writing; as you become involved in your subject matter, you may become enmeshed in the details and focus on learning or simply telling the information you have found. You need to do more than just repeat what you have read. Your writing should have a point, and you should be able to say it in a sentence. Sometimes instructors call this sentence a “thesis” or a “claim.”

So, if your instructor tells you to write about some aspect of oral hygiene, you do not want to just list: “First, you brush your teeth with a soft brush and some peanut butter. Then, you floss with unwaxed, bologna-flavored string. Finally, gargle with bourbon.” Instead, you could say, “Of all the oral cleaning methods, sandblasting removes the most plaque. Therefore it should be recommended by the American Dental Association.” Or, “From an aesthetic perspective, moldy teeth can be quite charming. However, their joys are short-lived.”

Convincing the reader of your argument is the goal of academic writing. It doesn’t have to say “argument” anywhere in the assignment for you to need one. Look at the assignment and think about what kind of argument you could make about it instead of just seeing it as a checklist of information you have to present. For help with understanding the role of argument in academic writing, see our handout on argument .

What kind of evidence do you need?

There are many kinds of evidence, and what type of evidence will work for your assignment can depend on several factors–the discipline, the parameters of the assignment, and your instructor’s preference. Should you use statistics? Historical examples? Do you need to conduct your own experiment? Can you rely on personal experience? See our handout on evidence for suggestions on how to use evidence appropriately.

Make sure you are clear about this part of the assignment, because your use of evidence will be crucial in writing a successful paper. You are not just learning how to argue; you are learning how to argue with specific types of materials and ideas. Ask your instructor what counts as acceptable evidence. You can also ask a librarian for help. No matter what kind of evidence you use, be sure to cite it correctly—see the UNC Libraries citation tutorial .

You cannot always tell from the assignment just what sort of writing style your instructor expects. The instructor may be really laid back in class but still expect you to sound formal in writing. Or the instructor may be fairly formal in class and ask you to write a reflection paper where you need to use “I” and speak from your own experience.

Try to avoid false associations of a particular field with a style (“art historians like wacky creativity,” or “political scientists are boring and just give facts”) and look instead to the types of readings you have been given in class. No one expects you to write like Plato—just use the readings as a guide for what is standard or preferable to your instructor. When in doubt, ask your instructor about the level of formality she or he expects.

No matter what field you are writing for or what facts you are including, if you do not write so that your reader can understand your main idea, you have wasted your time. So make clarity your main goal. For specific help with style, see our handout on style .

Technical details about the assignment

The technical information you are given in an assignment always seems like the easy part. This section can actually give you lots of little hints about approaching the task. Find out if elements such as page length and citation format (see the UNC Libraries citation tutorial ) are negotiable. Some professors do not have strong preferences as long as you are consistent and fully answer the assignment. Some professors are very specific and will deduct big points for deviations.

Usually, the page length tells you something important: The instructor thinks the size of the paper is appropriate to the assignment’s parameters. In plain English, your instructor is telling you how many pages it should take for you to answer the question as fully as you are expected to. So if an assignment is two pages long, you cannot pad your paper with examples or reword your main idea several times. Hit your one point early, defend it with the clearest example, and finish quickly. If an assignment is ten pages long, you can be more complex in your main points and examples—and if you can only produce five pages for that assignment, you need to see someone for help—as soon as possible.

Tricks that don’t work

Your instructors are not fooled when you:

  • spend more time on the cover page than the essay —graphics, cool binders, and cute titles are no replacement for a well-written paper.
  • use huge fonts, wide margins, or extra spacing to pad the page length —these tricks are immediately obvious to the eye. Most instructors use the same word processor you do. They know what’s possible. Such tactics are especially damning when the instructor has a stack of 60 papers to grade and yours is the only one that low-flying airplane pilots could read.
  • use a paper from another class that covered “sort of similar” material . Again, the instructor has a particular task for you to fulfill in the assignment that usually relates to course material and lectures. Your other paper may not cover this material, and turning in the same paper for more than one course may constitute an Honor Code violation . Ask the instructor—it can’t hurt.
  • get all wacky and “creative” before you answer the question . Showing that you are able to think beyond the boundaries of a simple assignment can be good, but you must do what the assignment calls for first. Again, check with your instructor. A humorous tone can be refreshing for someone grading a stack of papers, but it will not get you a good grade if you have not fulfilled the task.

Critical reading of assignments leads to skills in other types of reading and writing. If you get good at figuring out what the real goals of assignments are, you are going to be better at understanding the goals of all of your classes and fields of study.

You may reproduce it for non-commercial use if you use the entire handout and attribute the source: The Writing Center, University of North Carolina at Chapel Hill

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Assignment Problem

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

  • First, prefix name of what this variable represents.
  • Second is the list of all the variables.
  • Third is the lower bound on this variable.
  • Fourth variable is the upper bound.
  • Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

  • Solving Blending Problem in Python using Gurobi
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The Assignment Model

The linear programming formulation of the assignment model is similar to the formulation of the transportation model, except all the supply values for each source equal one, and all the demand values at each destination equal one. Thus, our example is formulated as follows :

assignment model examples

This is a balanced assignment model. An unbalanced model exists when supply exceeds demand or demand exceeds supply.

assignment model examples

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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

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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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13 Chapter 13: Last Step of Four Step Modeling (Trip Assignment Models)

Chapter 13 is the last chapter of the book unpacking the last step of four-step travel demand modeling, i.e., trip assignment. This step determines which paths travelers choose for moving between each pair of zones. Additionally, this step can yield numerous results such as traffic volumes in different transportation corridors, the patterns of vehicular movements, total VMTs and VTTs in the network, and zone-to-zone travel costs. The identification of the heavily congested links is crucial for transportation planning and engineering practitioners. This chapter begins with some fundamental concepts, such as the link cost functions. Next, it presents some common and useful trip assignment methods with relevant examples. The methods covered in this chapter include all-or-nothing (AON), user equilibrium (UE), system optimum (SO), feedback loop between distribution and assignment (LDA), incremental increase assignment, capacity restrained assignment, and stochastic user equilibrium assignment.

Learning Objectives

Student Learning Outcomes

  •  Describe the reasons for performing trip assignment models in FSM and relate these models’ foundation through the cost-function concept.
  • Compare static and dynamic trip assignment models and infer the appropriateness of each model for different situations.
  • Explain Wardrop principles and relate them to traffic assignment algorithms.
  • Complete simple network traffic assignment models using static models such as the all-or-nothing and user equilibrium models.
  • Solve modal split analyses manually for small samples using the discrete choice modeling framework and multinominal logit models.

Prep/quiz/assessments

  • Explain what the link performance function is in trip assignment models and how it is related to link capacity.
  • Name a few static and dynamic traffic assignment models and discuss how different their rules or algorithms are.
  • How does stochastic decision-making on route choice affect the transportation level of service, and how it is incorporated into traffic assignment problems?
  • Name one extension of the all-or-nothing assignment model and explain how this extension improves the model results.

13.1 Introduction

in this chapter, we continue the discussion  about FSM and elaborate on  different methods of traffic assignment, which is the last step in the FSM model after trip generation, trip distribution, and modal split. The traffic assignment step, which is also called route assignment or route choice , simulates the choice of route selection from a set of alternatives between origin zone and the destination zone (Levinson et al., 2014). After the first three steps, we know the number of trips produced between each pair of zones and what portion of these trips are completed by different transportation modes. As the final step, we would be also interested in determining what routes or links within our study areas will likely be used. For instance, in a Regional Transportation Plan (RTP), we would be interested in determining how much shift or diversion in daily traffic happens if we introduce an additional transit line or extent a highway corridor (Levinson et al., 2014). Similar to trip distribution, the impedance function has an important role in route choice for travelers. Normally, the impedance function is related to travel cost or travel time. The longer the trip or the higher the cost, the larger the impedance for the trip along that path (Wang & Hofe, 2008).

The output from the last step of the FSM model can provide modelers with numerous valuable results. Such results can yield the planner an insight of good and bad characteristics of various preconceived plans. The results of trip assignment analysis can be:

  • The traffic flows in the transportation system andthe pattern of vehicular movements
  • Volume of traffic on network links
  • Travel costs between trip origins and destinations
  • Aggregated network measures, e.g. total flows of vehicles, total distance travelled by vehicles (VMT) , total vehicle travel time (VTT)
  • Zone-to-zone travel costs (travel time) for a given demand level
  • Obtaining modeled link flows and highlighting congested corridors
  • Analysis of turning movements for future intersection design
  • Finding out which O-D pairs have taken a particular link or path
  • Simulation of the individual’s choice for each pair of origins and destinations (Mathew & Rao, 2006)

13.2 Link Performance Function

One of the most important and fundamental concepts of the traffic assignment process is to build a link performance function . This function is usually used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow. While this function can take different forms such as linear, polynomial , exponential , and hyperbolic , there are different equations that represent the above-mentioned relationship. One of the most common functions is the link cost function that represents generalized travel costs (United States Bureau of Public Roads, 1964). This equation estimates travel time on a free-flow road (travel with speed limit) adding a function that increases travel time exponentially as the road gets more congested. The congestion can be represented by the road volume to capacity ratio (Meyer, 2016). However, throughout recent years transportation planners have realized that in many cases, the delay on the links is caused by delays on the intersection, an observation that is not encompassed by the link cost function. Nevertheless, in the following sections we will resort to the traditional function. Equation (1) is the most common and general formula for link performance function.

t=t_o[1+\alpha\left(\frac{x}{k}\right)\beta]

  • t and x are the travel time and vehicle flow;
  • t 0 is the link free flow travel time;
  • k is the link capacity;
  • α and β are parameters for specific type of links and calibrated using the field data. In the absence of any field data, it is usually assumed = 0.15, and β= 4.0.

α and β are the coefficient for this formula and can take different values (model parameters). However, most studies and planning practices use the same value for them. That said, these values can be locally calibrated for the most efficient results. Also, Figure 13.1 shows the relationship between capacity and travel time. In this plot, the travel time remains constant by increase in vehicle volumes until the turning point which indicates the volume on the link is beginning to exceed the capacity.

This figure shows the exponential relationship between travel time and flow of traffic,

Figure 13.1 Link Flow and Travel Time Relationship

The following example shows how the link performance function helps us to determine the travel time according to flow and capacity.

13.2.1 Example 1

Assume the traffic volume on a path between zone i and j was 525. The travel time recorded on this path is 15 minutes. If the capacity of this path would be 550, then calculate the new travel time for future iteration of the model.

Based on the link performance function, we have:

Now we have to plug in the numbers into the formula to determine the new travel time:

t=15[1+\0.15\left(\frac{525}{550}\right)\4]=16.86

13.3 Traffic Assignment Models

Through the rest of this chapter, we are going to discuss different traffic assignment models. In general, the process of traffic assignment is usually done separately for private cars and transit systems. As we specified in previous chapters, the transit impedance function is different from private auto; thus, simulating a utility maximization behavior for a driver and rider should be different. For public transit assignment, variables such as fare, stop or transfer, waiting time, and trip times define the utility (equilibrium) (Sheffi, 1985). However, in some cases the two mentioned networks are related when public buses share highways with cars, and congestion can also affect the performance of public transit (Rojo, 2020). Typically, private car traffic assignment models the path choice of trip makers using:

  • algorithms like all-or-nothing
  • incremental
  • capacity-restrained
  • user equilibrium
  • system optimum assignment

User equilibrium is based on the principle assumption that travelers try to minimize their travel costs. In this algorithm, the equilibrium occurs when there is no user able to reduce their travel time or cost by changing path. This is the most popular algorithms employed for simulation in the U.S. (Meyer, 2016). Moreover, more recent trip assignment models use approaches such as:

  • static user-equilibrium assignment algorithm
  • “multiple-time-period assignment for multiple classes (for example, drive-alone, rideshare, and bike/walk)
  • an iterative feedback loop mechanism between, at a minimum, the network assignment step and the trip distribution step
  • separate specification of facilities like HOV and high-occupancy toll (HOT) lanes
  • independent transit assignment using congested highway travel times to estimate a bus ridership assignment” (Meyer, 2016, p.226).

13.3.1 All-or-nothing Model

Through the all-or-nothing (AON) assignment, we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level of service, meaning that the traffic time is not affected by the traffic flow on the path. The only logic behind this model is that each traveler simply uses the shortest path from his or her origin to the destination and no vehicle is assigned to other paths (Hui, 2014). This method is called the all-or-nothing assignment model and is the simplest one among all assignment models. This method is also called the 0-1 assignment model, and its advantage is its simple procedure and calculation. The assumptions of this method are:

  • Congestion does not affect travel time or cost, meaning that no matter how much traffic is loaded on the route, congestion does not take place.
  • Since the method assigns one route to any travel between each pair of OD, travelers traveling from particular zone to another particular zone all choose the same route (Hui, 2014).

To run the AON model, the following process can be followed:

  • Step 0: Initialization. Use free flow travel costs Ca=Ca(0) , for each link a on the empty network. Ɐ
  • Step 1: Path finding. Find the shortest path P for each zonal pair.
  • Step 2: Path flows assigning. Assign both passenger trips (hppod) and freight trips (hfpod) in PCEs from zonal o to d to path P.
  • Step 3: Link flows computing. Sum the flows on all paths going through a link as total flows of this link.

Example 2 illustrates the above-mentioned process for the AON model.

13.3.2 Example 2

Table 13.1 shows a trip distribution matrix with 4 zones. Using the travel costs between each pair of them shown in Figure 13.2, assign the traffic to the network.

Load the vehicle trips from the trip distribution table shown below using the AON technique. After assigning the traffic, illustrate the links and the traffic volume on each on them.

Table 13.1 Trip Distribution Results

This photo shows the hypothetical network and travel time between zones: 1-2: 5 mins 1-4: 10 min 4-2: 4 mins 3-2: 4 mins 3-4: 9 mins

Figure 13.2 Transportation Network

To solve this problem, we need to find the shortest path among all alternatives for each pair of zones. The result of this procedure would be 10 routes in total, each of which bears a specific amount of travels. For instance, the shortest path between zone 1 and 2 is the straight line with 5 min travel time. All other routes like 1 to 4 to 2 or 1 to 4 to 3 to 2 would be empty from travelers going from zone 1 to zone 2. The results are shown in Table 13.2.

Table 13.2 Traffic Volumes for Each Route

As you can see, some of the routes remained unused. This is because in all-or-nothing if a route has longer travel time or higher costs, then it is assumed it would not be used at all.

13.3.4 User Equilibrium

The next method for traffic assignment is called user equilibrium (UE). The rule or algorithm is adopted from the well- known Wardrop equilibrium (19 52) conditions (Correa & Stier-Moses, 2011). In this algorithm, it is assumed that travelers will always choose the shortest path and equilibrium condition would be realized when no traveler is able to decrease their travel impedance by changing paths (Levinson et al., 2014).

As we discussed, the UE method is based on the first principle of Wardrop : “for each origin-destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path”(Jeihani Koohbanani, 2004). The mathematical format of this principle is shown in equation (3):

T 1 =T 2       (2)

For a given OD pair, the UE condition can be expressed in equation (3):

fk\left(ck-u\right)=0:\forall k

This means that all paths will have the same travel time. Also, for this model we have the following general assumptions:

  • The users possess all the knowledge needed about different paths.
  • The users have perfect knowledge of the path cost.
  • Travel time in a route is subject to change only by the cost flow function of that route.
  • Travel times increases as we load travel into the network (Mathew & Rao, 2006).

Hence, the UE assignment comes to an optimization problem that can be formulated using equation (4):

Minimize\ Z=\sum_{a}\int_{0}^{Xa}ta\left(xa\right)dx

k  is the path x a equilibrium flow in link a t a  travel time on link a f k rs  flow on path  connecting OD pairs q rs  trip rate between  and δ a, k rs is constraint function defined as 1 if link a belongs to path k and 0 otherwise

Example 3 shows how the UE method can be applied for the traffic assignment step. This example is a very simple network consisting of two zones with two possible paths between them.

13.3.5 Example 3

This photo shows the hypothetical network with two possible paths between two zones 1: 5=4x_1 2: 3+2x_2 (to power of two)

Figure 13.3 A Simple Two-Zone System with Cost Function

In this example, t 1 and t 2 are travel times measured by min on each route, and x 1 and x 2 are traffic flows on each route measured by (Veh/Hour).

Using the UE method, assign 4,500 Veh/Hour to the network and calculate travel time on each route after assignment, traffic volume, and system total travel time.

According to the information provided, total flow (X 1 +X 2 ) is equal to 4,500 (4.5).

First, we need to check, with all traffic assigned to one route, whether that route is still the shortest path. Thus we have:

T 1 (4.5)=23min

T 2 (0)=3min

if all traffic is assigned to route 2:

T 1 (0)=3min

T 2 (4.5)=43.5 min

Step 2: Wardrope equilibrium rule: t 1 =t 2        5+4x 1 =3+ 2x 2 2         and we have x 1 =4.5-x 2

Now the equilibrium equation can be written as: 6 + 4(4.5 − x2)=4+ x222

x 1 = 4.5 − x 2 = 1.58

Now the updated average travel times are: t 1 =5+4(1.58)=11.3min and T 2 =3+2(2.92)2=20.05min

Now the total system travel time is:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=2920 veh/hr(11.32)+1585 veh/hr(20.05)=33054+31779=64833 min

13.3.6 System Optimum Assignment

One other traffic assignment model similar to the previous one is called system optimum (SO) in which the second principle of the Wardrop defines the logic of the model. Based on this principle, drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time (Mathew & Rao, 2006). Using the SO traffic assignment, problems like optimizing departure time for a single commuting route, minimizing total travels from multiple origins to one destination, or minimizing travel time in stochastic time-dependent OD flows from several origins to a single destination can be solved (Jeihani Koohbanani, 2004).

The basic mathematical formula for this model that satisfies the principle of the model is shown in equation (5):

minimize\ Z=\sum_{a}{xata\left(xa\right)}

In example 4, we will use the same network we described in the UE example in order to compare the results for the two models.

13.3.7 Example 4

In that simple two-zone network, we had:

T 1 =5+4X 1    T2=3+2X 2 2

Now, based on the principle of the model we have:

Z(x)=x 1 t 1 (x 1 )+x 2 t 2 (x 2 )

Z(x)=x 1 (5+4x 1 )+x 2 (3+2x 2 2 )

Z(x)=5x 1 +4x 1 2 +3x 2 +2x 2 3

From the flow conservation. we have: x 1 +x 2 =4.5     x 1 =4.5-x 2

Z(x)=5(4.5-x 2 )+4(4.5-x 2 )2+4x 2 +x 2 3

Z(x)=x 3 2 +4x 2 2 -27x 2 +103.5

In order to minimize the above equation, we have to take derivatives and equate it to zero. After doing the calculations, we have:

Based on our finding, the system travel time would be:

T 1 =5+4*1.94=12.76min     T 2 =3+ 2(2.56)2=10.52 min

And the total travel time of the system would be:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=1940 veh/hr(12.76)+2560 veh/hr(10.52)=24754+26931=51685 min

13.3.8 Feedback Loop Model (Combined Traffic Assignment and Trip Distribution)

In the feedback loop model , an interaction between the trip distribution route choice step with several iterations is defined. The essence of this model is that in case a route between an origin and destination gets too congested, then the traveler may replace their destination, for instance choosing between several shopping malls available in a region. In other words, in a real-world situation, travelers usually make decisions about their travel characteristics simultaneously (Qasim, 2012).

The chart below shows how the combination of these two modes can take place:

This photo shows the feedback loop in FSM.

Figure 13.4 The Feedback Loop between the Second and the Fourth Step

Equation (6), shown below for this model, ensures convergence at the end of the model is:

Min\funcapply\sum_a\hairsp\int_0^{p_a+f_a}\hairsp C_a(x)dx+\frac{1}{\zeta}\sum_o\hairsp\sum_d\hairsp T^{od}\left(\ln\funcapply T^{od}-K\right)

where C a (t) is the same as previous

P a , is total personal trip flows on link a,

f a ; is total freight trip flows on link a,

T od is the total flow from node o to node d,

p od is personal trip from node o to node d,

F od is freight trip from node o to node d,

ζ is a parameter estimated from empirical data,

K is a parameter depending on the type of gravity model used to calculate T od , Evans (1976) proved that K’ equals to 1 for distribution using doubly constrained gravity model and it equals to 1 plus attractiveness for distribution using singly constrained model. Florian et al. (1975) ignored K for distribution using a doubly constrained gravity model because it is a constant.

13.3.9 Incremental Increase model

Another model of traffic assignment we are going to elaborate on here is called incremental increase . In this model, which is based on the logic of the AON model, a process is designed with multiple steps. In each step or level, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume. Through this incremental addition of traffic, the travel time of each route in step (n) is the updated travel time from the previous step (n-1)(Rojo, 2020).

The steps for the incremental increase traffic assignment model are:

  • Finding the shortest path between each pair of O-Ds
  • Assigning a portion of the trips according to the matrix (usually 40, 30, 20 and 10 percent to the shortest path)
  • Updating the travel time after each iteration (each incremental increase)
  • Continuing until all trips are assigned
  • Summing the results

Example 4 illustrates the process of this method’s implementation.

13.3.10 Example 4

A hypothetical network accommodates two zones with three possible links between them. Perform an incremental increase traffic assignment model for assigning 200 trips between the two zones with increments of: 30%, 30%, 20%, 20%. (The capacity is 50 trips.)

This photo shows the hypothetical network with two possible paths between two zones 1: 6 mins 2: 7 mins 3: 12 mins

Figure 13.5 A Two-Zone Network with Three Possible Routes

Step 1 (first iteration): Using the method of AON, we now assign the flow to the network using the function below:

t=to[1+\alpha\left(\frac{x}{k}\right)\beta]

Since the first route has the shortest travel time, the first 30% of the trips will be assigned to route 1. The updated travel time for this path would be:

t=6\left[1+0.15\left(\frac{60}{50}\right)4\right]=7.86

And the remaining route will be empty, and thus their travel times are unchanged.

Step 2 (second iteration): Now, we can see that the second route has the shortest travel time, with 30% of the trips being assigned to this route, and the new travel time would be:

t=7\left[1+0.15\left(\frac{60}{50}\right)4\right]=9.17

Step 3 (third iteration): In the third step, the 20% of the remaining trips will be assigned to the shortest path, which in this case is the first route again. The updated travel time for this route is:

t=7.86\left[1+0.15\left(\frac{40}{50}\right)4\right]=8.34

Step 4 (fourth iteration): In the last iteration, the remaining 10% would be assigned to first route, and the time is:

t=8.34\left[1+0.15\left(\frac{40}{50}\right)4\right]=8.85

Finally, we can see that route 1 has a total of 140 trips with a 8.85 travel time, the second route has a total of 60 trips with a 9.17 travel time, and the third route was never used.

13.3.11 Capacity Restraint Assignment

Thus far, in all the algorithms or rules presented, the capacity of the link was incorporated into the model, and travel time was the only factor for assigning the flow to a link. In this model, after each iteration, the total number of trips are compared with the capacity to observe how much increase in travel time was realized by the added volume. In this model, the iteration stops if the added volume in step (n) does not change the travel time updated in step (n-1). With the incorporation of such a constraint, the cost or performance function would be different from cost functions discussed in previous algorithms (Mathew & Rao, 2006). Figure 13.6 visualizes the relationship between flow and travel time with a capacity constraint.

This figure shows the exponential relationship between travel time and flow of traffic with capacity line.

Figure 13.6 Link Flow and Travel Time Relationship

Based on this capacity constraint specific to each link, the α, β can be readjusted for different links such as highways, freeways, and other roads.

13.3.12 Stochastic User Equilibrium Traffic Assignment

Stochastic user equilibrium traffic assignment is a sophisticated and more realistic model, in which the level of uncertainty in regard to which link should be used based on a measurement of utility function is introduced. This model performs a discrete choice analysis through a logistic model. In this model, it is assumed that based on the first Wardrop principle, all drivers perceive the costs of traveling in each link identically and choose the route with minimum cost. In stochastic UE, however, the model allows different individuals to have different perceptions about the costs, and thus, they may choose non-minimum cost routes as well( Mathew & Rao, 2006). In this model, the flow is assigned to almost all links from the beginning (unlike previous models) which is closer to reality. The probability of using each path is calculated with the following logit formula shown in equation (7):

Pi=\frac{e^{ui}}{\sum_{i=1}^{k}e^{ui}}

P i is the probability of using path i

U i is the utility function for path i

In the following, an example of a simple network is presented.

13.3.13 Example 6

There is a flow of 200 trips between two points and their possible path, each of which has a travel time specified in Figure 13.7.

This photo shows the hypothetical network with two possible paths between two zones 1: 21 mins 2: 23 mins 3: 26 mins

Figure 13.7 A Simple Two-Zone Network with Three Links

Using the mentioned logit formula for these paths, we have:

P1=\frac{e^{-21i}}{e^{-21i}+e^{-23}+e^{-26i}}=0.875

Based on the calculated probabilities, the distribution of the traffic flow would be:

Q 1 =175 trips

Q 2 =24 trips

Q 3 =1 trips

13.3.14 Dynamic Traffic Assignment

Recall the first Wardrop principle, in which travelers are believed to choose their routes with the minimum cost. Dynamic traffic assignment is based on the same rule, but the difference is that delays resulted from congestion. In this way, not only travelers’ route choice affects the network’s level of service, but also the network’s level of service affects travelers’ choice as well. However, it is not theoretically proven that an equilibrium would result under such conditions (Mathew & Rao, 2006).

Today, various algorithms are developed to solve traffic assignment problems. In any urban transportation system, travelers’ route choice and different links’ level of service have a dynamic feedback loop and affect each other simultaneously. However, a lot of these rules are not present in the models presented here. In real world cases, there can be more than thousands of nodes and links in the network, and therefore more sensitivity to dynamic changes is required for a realistic traffic assignment (Meyer, 2016). Also, the travel demand model applies a linear sequence of the four steps, in which case it is also unlike reality. In fact, travelers may have narrow knowledge about all possible paths, modes, and opportunities and may not make rational decisions.

Route choice is the process of choosing a certain path for a trip from a very large choice sets.

Regional Transportation Plan is long term planning document for a region’s transportation usually updated every five years.

Vehicles (VMT) is the aggregate number of miles deriven from in an area in particular time of day.

Total vehicle travel time is the aggregate amount of time spent in transportation usually in minutes.

Link performance function is function used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow.

Hyperbolic function is a function used for linear differential equations like calculating distances and angels in hyperbolic geometry.

Free-flow road is situation where vehicles can travel with the maximum allowed travel speed.

  • Algorithms like all-or-nothing an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level of service, meaning that the traffic time is not affected by the traffic flow on the path.

Capacity-restrained is a model which takes into account the capacity of a road compared to volume and updates travel times.

User equilibrium is a traffic assignment model where we assume that travelers will always choose the shortest path and equilibrium condition would be realized when no traveler is able to decrease their travel impedance by changing paths.

System optimum assignment is an assignment model based on the principle that drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time.

Static user-equilibrium assignment algorithm is an iterative traffic assignment process which assumes that travelers chooses the travel path with minimum travel time subject to constraints.

  • Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination.
  • First principle of Wardrop is the assumption that for each origin-destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path.
  • System optimum (SO) is a condition in trip assignment model where total travel time for the whole area is at a minimum.
  • Stochastic time-dependent OD is a modeling framework where generation and distribution of trips are randomly assigned to the area.
  • Incremental increase is AON-based model with multiple steps in each of which, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume.
  • Stochastic user equilibrium traffic assignment employs a probability distribution function that controls for uncertainties when drivers compare alternative routes and make decisions.
  • Dynamic traffic assignment is a model based on Wardrop first principle in which delays resulted from congestion is incorporated in the algorithm.

Key Takeaways

In this chapter, we covered:

  • Traffic assignment is the last step of FSM, and the link cost function is a fundamental concept for traffic assignment.
  • Different static and dynamic assignments and how to perform them using a simplistic transportation network.
  • Incorporating stochastic decision-making about route choice and how to solve assignment problems with regard to this feature.

Correa, J.R., & Stier-Moses, N.E.(2010).Wardrope equilibria. In J.J. Cochran( Ed.), Wiley encyclopedia of operations research and management science (pp.1–12). Hoboken, NJ: John Wiley & Sons. http://dii.uchile.cl/~jcorrea/papers/Chapters/CS2010.pdf

Hui, C. (2014). Application study of all-or-nothing assignment method for determination of logistic transport route in urban planning. Computer Modelling & New Technologies , 18 , 932–937. http://www.cmnt.lv/upload-files/ns_25crt_170vr.pdf

Jeihani Koohbanani, M. (2004).  Enhancements to transportation analysis and simulation systems (Unpublished Doctoral dissertation, Virginia Tech). https://vtechworks.lib.vt.edu/bitstream/handle/10919/30092/dissertation-final.pdf?sequence=1&isAllowed=y

Levinson, D., Liu, H., Garrison, W., Hickman, M., Danczyk, A., Corbett, M., & Dixon, K. (2014). Fundamentals of transportation . Wikimedia. https://upload.wikimedia.org/wikipedia/commons/7/79/Fundamentals_of_Transportation.pdf

Mathew, T. V., & Rao, K. K. (2006). Introduction to transportation engineering. Civil engineering–Transportation engineering. IIT Bombay, NPTEL ONLINE, Http://Www. Cdeep. Iitb. Ac. in/Nptel/Civil% 20Engineering .

Meyer, M. D. (2016). Transportation planning handbook . John Wiley & Sons.

Qasim, G. (2015). Travel demand modeling: AL-Amarah city as a case study . [Unpublished Doctoral dissertation , the Engineering College University of Baghdad]

Rojo, M. (2020). Evaluation of traffic assignment models through simulation. Sustainability , 12 (14), 5536. https://doi.org/10.3390/su12145536

Sheffi, Y. (1985). Urban transportation networks: Equilibrium analysis with mathematical programming method . Prentice-Hall. http://web.mit.edu/sheffi/www/selectedMedia/sheffi_urban_trans_networks.pdf

US Bureau of Public Roads.  (1964). Traffic assignment manual for application with a large, high speed computer . U.S. Department of Commerce, Bureau of Public Roads, Office of Planning, Urban Planning Division.

https://books.google.com/books/about/Traffic_Assignment_Manual_for_Applicatio.html?id=gkNZAAAAMAAJ

Wang, X., & Hofe, R. (2008). Research methods in urban and regional planning . Springer Science & Business Media.

Polynomial is distribution that involves the non-negative integer powers of a variable.

Hyperbolic function is a function that the uses the variable values as the power to the constant of e.

A point on the curve where the derivation of the function becomes either maximum or minimum.

all-or-nothing is an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level

Incremental model is a model that the predictions or estimates or fed into the model for forecasting incrementally to account for changes that may occur during each increment.

Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination

feedback loop model is type of dynamic traffic assignment model where an iteration between route choice and traffic assignment step is peformed, based on the assumption that if a particular route gets heavily congested, the travel may change the destination (like another shopping center).

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Hungarian Method Examples

Now we will examine a few highly simplified illustrations of Hungarian Method for solving an assignment problem .

Later in the chapter, you will find more practical versions of assignment models like Crew assignment problem , Travelling salesman problem , etc.

Example-1, Example-2

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.

This is a minimization example of assignment problem . We will use the Hungarian Algorithm to solve this problem.

Identify the minimum element in each row and subtract it from every element of that row. The result is shown in the following table.

"A man has one hundred dollars and you leave him with two dollars, that's subtraction." -Mae West

On small screens, scroll horizontally to view full calculation

Identify the minimum element in each column and subtract it from every element of that column.

Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way:

  • For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.
  • If for a row and a column, there are two or more zeros and one cannot be chosen by inspection, choose the cell arbitrarily for assignment.

An optimal assignment is found, if the number of assigned cells equals the number of rows (and columns). In case you have chosen a zero cell arbitrarily, there may be alternate optimal solutions. If no optimal solution is found, go to step 5.

Use Horizontal Scrollbar to View Full Table Calculation

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix obtained from step 3 by adopting the following procedure:

  • Mark all the rows that do not have assignments.
  • Mark all the columns (not already marked) which have zeros in the marked rows.
  • Mark all the rows (not already marked) that have assignments in marked columns.
  • Repeat steps 5 (ii) and (iii) until no more rows or columns can be marked.
  • Draw straight lines through all unmarked rows and marked columns.

You can also draw the minimum number of lines by inspection.

Select the smallest element (i.e., 1) from all the uncovered elements. Subtract this smallest 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.

Now again make the assignments for the reduced matrix.

Final Table: Hungarian Method

Since the number of assignments is equal to the number of rows (& columns), this is the optimal solution.

The total cost of assignment = A1 + B4 + C2 + D3

Substituting values from original table: 20 + 17 + 17 + 24 = Rs. 78.

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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

assignment model examples

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

assignment model examples

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

assignment model examples

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

assignment model examples

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

Step 3 (Assignment):

assignment model examples

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

assignment model examples

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

assignment model examples

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

assignment model examples

Column 3 contains no zero. Go to Step 2.

assignment model examples

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

assignment model examples

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

assignment model examples

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

assignment model examples

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

assignment model examples

Step 3 (Assignment) :

assignment model examples

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

assignment model examples

The optimal assignment (minimum) cost = ₹ 35

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Assignment Method: Examples of How Resources Are Allocated

assignment model examples

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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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming.   The simplex method of Linear Programming Problems(LPP)   proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point.   Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres.     4 x 5 = 20 routes are possible.   Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs?   The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person?   The objective is minimizing total cost.   This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above.   Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

  • To decide the transportation of new materials from various centres to different manufacturing plants.   In the case of multi-plant company this is highly useful.
  • To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.
  • To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.   These two are the uses of transportation model.   The objective is minimizing transportation cost.

Assignment model is used in the following:

  • To decide the assignment of jobs to persons/machines, the assignment model is used.
  • To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
  • To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand.   Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available.   In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

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COMMENTS

  1. Assignment Model

    What is Assignment Model? → Assignment model is a special application of Linear Programming Problem (LPP), in which the main objective is to assign the work or task to a group of individuals such that; i) There is only one assignment.

  2. 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.

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    Assignment model - Example 1 (Balanced) maxus knowledge 27K subscribers Subscribe Subscribed 114 Share 45K views 10 years ago Assignment model In this video you will learn how to solve an...

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    Worked example of assigning tasks to an unequal number of workers using the Hungarian method The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks.

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    This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver. Example In the example there are five workers (numbered 0-4)...

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    is right or wrong, so write the assignment in whichever order feels best for you. The introduction might be up to around 10% of the word count (e.g. up to 200 words for a 2000 word assignment). Don't forget your conclusion At the end of the assignment, you need to summarise the key points you've made. You won't be introducing

  7. Operations Research with R

    For the following example, let's consider the following mathematical model to be solved: LP Assignment Problem — Example Let's take a look at the R code! Assignment Problem R Code Solution: LP Assignment Problem — Solution Total assignment cost: $27

  8. Understanding Assignments

    For example: "Throughout history, gerbils have played a key role in politics," or "In the last few weeks of class, we have focused on the evening wear of the housefly …" The Task of the Assignment Pay attention; this part tells you what to do when you write the paper. Look for the key verb or verbs in the sentence.

  9. Solving Assignment Problem using Linear Programming in Python

    Assignment Problem. A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming. For example, an operation manager needs to assign four jobs to four machines.

  10. The Assignment Model

    The following example will demonstrate the assignment model. The Atlantic Coast Conference (ACC) has four basketball games on a particular night. The conference office wants to assign four teams of officials to the four games in a way that will minimize the total distance traveled by the officials.

  11. Assignment model

    In this video, you will learn how to solve an unbalanced assignment model problem.

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    In this video you will learn how to solve an assignment model problem for with maximization objective

  13. What is 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 ... Get Quantitative Techniques: Theory and Problems now with the O'Reilly learning platform.

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    Example : Joe Klyne's three children, John, Karen, and Terri, want to earn some money to take care of personal expenses during a schoo! trip to the local zoo. Mr. Klyne has chosen three chores for his children: mowing the lawn, painting the garage door, and washing the family cars.

  15. Chapter 13: Last Step of Four Step Modeling (Trip Assignment Models

    Abstract: Chapter 13 is the last chapter of the book unpacking the last step of four-step travel demand modeling, i.e., trip assignment. This step determines which paths travelers choose for moving between each pair of zones.

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    The assignment problem deals with allocating various resources (items) to various activities (receivers) on a one to one basis, i.e., the number of operations are to be assigned to an equal number of operators where each operator performs only one operation. For example, suppose an accounts officer has 4 subordinates and 4 tasks.

  17. 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.

  18. Solution of assignment problems (Hungarian Method)

    Solution: 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.

  19. Assignment Method: Examples of How Resources Are Allocated

    Business Business Essentials Assignment Method: Examples of How Resources Are Allocated By Will Kenton Updated December 22, 2022 Reviewed by JeFreda R. Brown What Is the Assignment Method?...

  20. What are the examples of assignment model in operation research?

    Solution: The task is to assign 1 job to 1 person so that the total number of hours are minimized. By observation, we would simply assign the minimum time taken by a person to do the job. As per the observation: Task A should be assigned to I Task B should be assigned to III Task C should be assigned to IV Task D should be assigned to IV

  21. Transportation, Transshipment and Assignment Models

    Assignment Model Example 1 Fix-It Shop Example Received three new rush projects to repair: (1) a radio, (2) a toaster oven, and (3) a broken coffee table. Three workers (each has different talents and abilities). Estimated costs to assign each worker to each of the three projects. Assignment Model Example 1 Fix-It Shop • Rows denote people or ...

  22. Transportation and Assignment Models in Operations Research

    Assignment model is used in the following: To decide the assignment of jobs to persons/machines, the assignment model is used. To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).

  23. What Are Assignment Models

    The traffic assignment model is also used to generate the estimates of network performance that are used in the mode choice and trip distribution or destination choice stages of many models. Standard outputs of a traffic assignment are the link flows, volume-to-capacity ratios, congested travel times, and congested speeds.