what is a objective assignment problem

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

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That actually explain what's on your next test, assignment problem, from class:, optimization of systems.

The assignment problem is a type of optimization problem where the goal is to assign resources to tasks in the most efficient way possible. It typically involves a cost matrix that quantifies the cost associated with each potential assignment, and the objective is to minimize the total cost or maximize the overall effectiveness of the assignments. This problem is prevalent in various fields such as logistics, scheduling, and resource allocation.

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5 Must Know Facts For Your Next Test

• The assignment problem can be solved using various methods, including the Hungarian algorithm, which efficiently finds the optimal assignment in polynomial time.
• In its simplest form, the problem involves assigning 'n' tasks to 'n' agents while ensuring that each agent is assigned exactly one task and each task is assigned to exactly one agent.
• The assignment problem can be extended to accommodate constraints, such as maximum capacities or minimum requirements for certain assignments.
• Real-world applications of the assignment problem include job assignments in workforce management, matching students to schools, and routing deliveries in logistics.
• The Hungarian algorithm not only provides an optimal solution but also runs in O(n^3) time complexity, making it suitable for moderate-sized problems.

Review Questions

• The cost matrix serves as a foundational element in solving the assignment problem by providing a structured way to represent and quantify the costs associated with each potential resource-task pairing. Each element in this matrix indicates the cost of assigning a specific resource to a particular task. The optimization process then uses this matrix to identify the combination of assignments that results in the lowest total cost or maximizes efficiency. Understanding how to manipulate and interpret this matrix is crucial for finding optimal solutions effectively.
• The Hungarian algorithm is specifically designed for solving linear assignment problems efficiently and offers distinct advantages over other methods such as brute force or combinatorial approaches. Unlike these methods that may involve checking every possible combination of assignments, leading to exponential time complexity, the Hungarian algorithm operates in polynomial time (O(n^3)). This efficiency makes it particularly suitable for moderate-sized problems where quick solutions are needed. Furthermore, it guarantees an optimal solution by systematically adjusting potential costs until the most effective assignments are identified.
• Real-world scenarios such as workforce management or logistics can be effectively modeled as assignment problems by framing them in terms of resources needing to be assigned to tasks while considering associated costs. For instance, assigning delivery trucks to routes can minimize fuel costs and time while maximizing delivery efficiency. Solving these problems optimally has significant implications, including cost savings, improved service levels, and better resource utilization. Understanding how to frame real-life situations as mathematical problems allows for leveraging optimization techniques like the Hungarian algorithm to achieve tangible benefits across industries.

Related terms

Cost Matrix : A table that represents the costs associated with assigning different resources to various tasks, where each cell indicates the cost of a specific assignment.

Bipartite Graph : A graph that consists of two disjoint sets of vertices, where edges only connect vertices from one set to the other, often used to represent the relationships in assignment problems.

Optimal Solution : The best possible assignment configuration that minimizes costs or maximizes efficiency in an assignment problem, achieved through various optimization algorithms.

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• Mathematical Methods for Optimization

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Introduction, section snippets, references (72), cited by (408).

European Journal of Operational Research

Discrete optimization assignment problems: a golden anniversary survey, the classic assignment problem, models with multiple tasks per agent, multi-dimensional assignment problems, bottleneck assignment problems under categorization, computers & operations research, multiple bottleneck assignment problem, a heuristic procedure for the crew rostering problem, vehicle routing considerations in distribution system design, quadratic assignment problems, lexicographic bottleneck problems, operations research letters, development and evaluation of an assignment heuristic for allocating cross-trained workers, a multi-level bottleneck assignment approach to the bus drivers’ rostering problem, a survey of algorithms for the generalized assignment problem, the β-assignment problems, the k -cardinality assignment problem, discrete applied mathematics, minimum deviation and balanced optimization: a unified approach, slotmanager: a microcomputer-based decision support system for university timetabling, decision support systems, on an assignment problem with side constraints, computers & industrial engineering, a variation of the assignment problem, the three-dimensional assignment problem with capacity constraints, minimum deviation problems, tabu search for the multilevel generalized assignment problem, formulating and solving production planning problems, balanced optimization problems, the bottleneck generalized assignment problem, bottleneck generalized assignment problems, engineering costs and production economics, stable solutions vs. multiplicative utility solutions for the assignment problem, using the generalized assignment problem in scheduling the rosat space telescope, computer scheduling of medical school clerkships, computers & education, an integer programming model for the allocation of databases in a distributed computer system, an algorithm for fractional assignment problems, capacity planning by the dynamic multi-resource generalized assignment problem (dmrgap), lexicographic bottleneck combinatorial problems, an optimal solution to a dock door assignment problem, linear and semi-assignment problems: a core oriented approach, solving some lexicographic multi-objective combinatorial problems, a note on the assignment problem with seniority and job priority constraints, an introduction to timetabling, heuristic and exact algorithms for the simultaneous assignment problem, a variant of time minimizing assignment problem, employees recruitment: a prescriptive analytics approach via machine learning and mathematical programming, robust nurse-to-patient assignment in home care services to minimize overtimes under continuity of care, optimization for dynamic ride-sharing: a review, simultaneous task allocation and planning for temporal logic goals in heterogeneous multi-robot systems, a distributed version of the hungarian method for multirobot assignment, a partition-based match making algorithm for dynamic ridesharing.

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Assignment problem in linear programming : introduction and assignment model.

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Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Related Articles:

• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
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Assignment Problem: Maximization

There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.

The Hungarian Method can also solve such assignment problems , as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss.

The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.

• Unbalanced Assignment Problem
• Multiple Optimal Solutions

Example: Maximization In An Assignment Problem

At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.

How should the counters be assigned to persons so as to maximize the profit ?

Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.

On small screens, scroll horizontally to view full calculation

Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.

Select the smallest element from all the uncovered elements, i.e., 4. Subtract this element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment. Repeating step 3, we obtain a solution which is shown in the following table.

Final Table: Maximization Problem

Use Horizontal Scrollbar to View Full Table Calculation

The total cost of assignment = 1C + 2E + 3A + 4D + 5B

Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.

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An Assignment Problem and Its Application in Education Domain: A Review and Potential Path

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A Davidson College multi-objective assignment problem: a case study

• Published: 22 April 2014
• Volume 12 , pages 379–401, ( 2014 )

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• Timothy P. Chartier 1 ,
• Victoria Ellison 2 &
• Amy N. Langville 3  

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This paper presents a multi-objective linear integer program that assigns student volunteers to present lectures at participating classes in local schools. A student’s class assignment is based upon his or her availability to teach at that time as well as several additional factors including student preferences regarding commuting and partners as well as the institution’s goal of creating diverse student groups. This case study shows that the proposed mathematical program dramatically improves the assignments of students to classes and provides increased flexibility for modeling other goals and factors in future years. In addition, this multi-phase model can be applied in other contexts, such as crew scheduling or the scheduling of parallel sessions of large conferences.

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Acknowledgments

The authors are grateful to DASH Optimization for their Xpress-MP software, which was supplied through their generous Academic Partner Program.

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Timothy P. Chartier

Operations Research Program, N. C. State University, Raleigh, NC, 27695-8205, USA

Victoria Ellison

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Correspondence to Amy N. Langville .

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This research was supported in part by a research fellowship from the Alfred P. Sloan Foundation [T.C.P] and an NSF grant CISE-CCF-AF-1116963 [A.N.L].

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Chartier, T.P., Ellison, V. & Langville, A.N. A Davidson College multi-objective assignment problem: a case study. 4OR-Q J Oper Res 12 , 379–401 (2014). https://doi.org/10.1007/s10288-014-0259-2

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• How to Write a Problem Statement | Guide & Examples

How to Write a Problem Statement | Guide & Examples

Published on November 6, 2022 by Shona McCombes and Tegan George. Revised on November 20, 2023.

A problem statement is a concise and concrete summary of the research problem you seek to address. It should:

• Contextualize the problem. What do we already know?
• Describe the exact issue your research will address. What do we still need to know?
• Show the relevance of the problem. Why do we need to know more about this?
• Set the objectives of the research. What will you do to find out more?

Table of contents

When should you write a problem statement, step 1: contextualize the problem, step 2: show why it matters, step 3: set your aims and objectives.

Problem statement example

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Frequently asked questions about problem statements.

There are various situations in which you might have to write a problem statement.

In the business world, writing a problem statement is often the first step in kicking off an improvement project. In this case, the problem statement is usually a stand-alone document.

In academic research, writing a problem statement can help you contextualize and understand the significance of your research problem. It is often several paragraphs long, and serves as the basis for your research proposal . Alternatively, it can be condensed into just a few sentences in your introduction .

A problem statement looks different depending on whether you’re dealing with a practical, real-world problem or a theoretical issue. Regardless, all problem statements follow a similar process.

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The problem statement should frame your research problem, giving some background on what is already known.

Practical research problems

For practical research, focus on the concrete details of the situation:

• Where and when does the problem arise?
• Who does the problem affect?
• What attempts have been made to solve the problem?

Theoretical research problems

For theoretical research, think about the scientific, social, geographical and/or historical background:

• What is already known about the problem?
• Is the problem limited to a certain time period or geographical area?
• How has the problem been defined and debated in the scholarly literature?

The problem statement should also address the relevance of the research. Why is it important that the problem is addressed?

Don’t worry, this doesn’t mean you have to do something groundbreaking or world-changing. It’s more important that the problem is researchable, feasible, and clearly addresses a relevant issue in your field.

Practical research is directly relevant to a specific problem that affects an organization, institution, social group, or society more broadly. To make it clear why your research problem matters, you can ask yourself:

• What will happen if the problem is not solved?
• Who will feel the consequences?
• Does the problem have wider relevance? Are similar issues found in other contexts?

Sometimes theoretical issues have clear practical consequences, but sometimes their relevance is less immediately obvious. To identify why the problem matters, ask:

• How will resolving the problem advance understanding of the topic?
• What benefits will it have for future research?
• Does the problem have direct or indirect consequences for society?

Finally, the problem statement should frame how you intend to address the problem. Your goal here should not be to find a conclusive solution, but rather to propose more effective approaches to tackling or understanding it.

The research aim is the overall purpose of your research. It is generally written in the infinitive form:

• The aim of this study is to determine …
• This project aims to explore …
• This research aims to investigate …

The research objectives are the concrete steps you will take to achieve the aim:

• Qualitative methods will be used to identify …
• This work will use surveys to collect …
• Using statistical analysis, the research will measure …

The aims and objectives should lead directly to your research questions.

Learn how to formulate research questions

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You can use these steps to write your own problem statement, like the example below.

Step 1: Contextualize the problem A family-owned shoe manufacturer has been in business in New England for several generations, employing thousands of local workers in a variety of roles, from assembly to supply-chain to customer service and retail. Employee tenure in the past always had an upward trend, with the average employee staying at the company for 10+ years. However, in the past decade, the trend has reversed, with some employees lasting only a few months, and others leaving abruptly after many years.

Step 2: Show why it matters As the perceived loyalty of their employees has long been a source of pride for the company, they employed an outside consultant firm to see why there was so much turnover. The firm focused on the new hires, concluding that a rival shoe company located in the next town offered higher hourly wages and better “perks”, such as pizza parties. They claimed this was what was leading employees to switch. However, to gain a fuller understanding of why the turnover persists even after the consultant study, in-depth qualitative research focused on long-term employees is also needed. Focusing on why established workers leave can help develop a more telling reason why turnover is so high, rather than just due to salaries. It can also potentially identify points of change or conflict in the company’s culture that may cause workers to leave.

Step 3: Set your aims and objectives This project aims to better understand why established workers choose to leave the company. Qualitative methods such as surveys and interviews will be conducted comparing the views of those who have worked 10+ years at the company and chose to stay, compared with those who chose to leave.

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

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Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

Once you’ve decided on your research objectives , you need to explain them in your paper, at the end of your problem statement .

Keep your research objectives clear and concise, and use appropriate verbs to accurately convey the work that you will carry out for each one.

I will compare …

All research questions should be:

• Focused on a single problem or issue
• Researchable using primary and/or secondary sources
• Feasible to answer within the timeframe and practical constraints
• Specific enough to answer thoroughly
• Complex enough to develop the answer over the space of a paper or thesis
• Relevant to your field of study and/or society more broadly

Research objectives describe what you intend your research project to accomplish.

They summarize the approach and purpose of the project and help to focus your research.

Your objectives should appear in the introduction of your research paper , at the end of your problem statement .

Your research objectives indicate how you’ll try to address your research problem and should be specific:

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A Target-Assignment Problem

Alan S. Manne

Cowles Foundation for Research in Economics at Yale University, New Haven, Connecticut

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This paper is concerned with a target assignment model of a probabilistic and nonlinear nature, but nevertheless one which is closely related to the “personnel-assignment” problem. It is shown here that, despite the apparent nonlinearities, it is possible to devise a linear programming formulation that will ordinarily provide a close approximation to the original problem.

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• Analysis of a maximum marginal return assignment algorithm
• Recent Developments in Linear Programming
• An approximating algorithm for an optimum aim-points problem Naval Research Logistics Quarterly, Vol. 7, No. 2

Volume 6, Issue 3

Article information.

• Published Online: June 01, 1958

© 1958 INFORMS

The objective of an assignment problem is to assign ______. - Mathematics and Statistics

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The objective of an assignment problem is to assign ______.

Number of jobs to equal number of persons at maximum cost.

Number of jobs to equal number of persons at minimum cost

Only the maximize cost

Only to minimize cost

Solution Show Solution

The objective of an assignment problem is to assign number of jobs to equal number of persons at minimum cost .

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