assignment problem definition

Assignment Problem: Meaning, Methods and Variations | Operations Research

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

Meaning of Assignment Problem:

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

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

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

Definition of Assignment Problem:

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

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

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Assignment problem

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

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

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

(origins or supply),

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

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

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

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

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

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

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

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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.

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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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

• Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)
• Introduction to Exact Cover Problem and Algorithm X
• Greedy Approximate Algorithm for Set Cover Problem
• Job Assignment Problem using Branch And Bound
• Implementation of Exhaustive Search Algorithm for Set Packing
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• Transportation Problem | Set 1 (Introduction)
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• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

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Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
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• Phases and Processes of O.R. Study
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2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
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4 Transportation Problem

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

5 Assignment Problem

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

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
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• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
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8 Integer Programming

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9 Dynamic Programming

• Dynamic Programming Methodology: An Example
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10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
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11 Introduction to game theory and its Applications

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12 Monte Carlo Simulation

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assignment problem

(classic problem)

Definition: The problem of finding a maximum (or minimum) weight matching in a weighted , bipartite graph .

Also known as marriage problem.

See also Munkres' assignment algorithm .

Note: From Algorithms and Theory of Computation Handbook, page 7-21, Copyright © 1999 by CRC Press LLC. Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC.

Author: CRC-A

If you have suggestions, corrections, or comments, please get in touch with Paul Black .

Entry modified 16 May 2005. HTML page formatted Wed Mar 13 12:42:45 2019.

Cite this as: Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "assignment problem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 16 May 2005. (accessed TODAY) Available from: https://www.nist.gov/dads/HTML/assignment.html

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)

• 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

→ 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.

Suggested Notes:

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Indirect cost less than Crash Cost

Basics of Program Evaluation and Review Technique (PERT)

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Graphical Method | Methods to solve LPP | Linear Programming

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Linear Programming Problem (LPP) Formulation with Numericals

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Why Defining Exactly Who Is and Isn’t an Artist Matters

Official definitions of 'artist' (of which there are a confusing many) come into play when artists are counted in a census, have to pay income taxes or want to apply for grants and live-work spaces..

The question of how to define art has plagued creatives and philosophers for centuries. Aristotle defined art as a true idea given physical form. Leo Tolstoy called it “a means of union among men, joining them together in the same feelings, and indispensable for the life and progress toward well-being of individuals and of humanity,” and Oscar Wilde identified art as “the most intense mode of individualism the world has known.” It seems we can, individually, define art, but we can’t reach a consensus.

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The same issue arises when the goal is to define ‘artist.’ Lots of people want to be viewed as artists, and label themselves such, from tattooists (body artists) to chefs (culinary artists) to, more recently, GPT prompters (A.I. artists). If nailing down the definition of artist is a semantics issue, it’s also one with real-world consequences. In 2023, a Colorado web designer successfully claimed before the U.S. Supreme Court that she was an artist—versus a mere service provider—which meant she was exempt from the state’s public accommodations law and could refuse to create wedding websites for same-sex couples.

SEE ALSO: The Costume Institute’s ‘Sleeping Beauties: Reawakening Fashion’ Is Full of Couture Corpses

In fact, there are numerous spheres in which defining what an artist is and isn’t is important. There are, for instance, grants and studio or residential spaces set aside exclusively for artists, and the organizations or government agencies overseeing their assignment need to be clear on who is and isn’t an artist. Then there are surveys conducted by economic, social and cultural researchers into artists’ employment, artists’ healthcare coverage and needs, the economic benefits of creative communities and other related inquiries for whom broader definitions of artist create a methodological problem. Without the licenses, permits, state testing or reported income requirements of other professionals, determining who is or isn’t an artist starts to feel like a value judgment.

That said, sometimes proving that someone is an artist is not just about cultural cachet or bragging rights—”official” definitions of artist (of which there are a confusing many) come into play when artists are counted in a census or have to pay their income taxes. In these cases, a self-proclaimed artist might have to meet several criteria dictated by the U.S. government to show that they’re also an artist by trade.

An artist’s job is art . The Bureau of the Census (whose data is used by the Bureau of Labor Statistics, the National Endowment for the Arts and other agencies) makes a broad national survey every ten years, inquiring about sources of paid employment during the census week. People with more than one source of income are counted occupationally in the job at which they worked the greatest number of hours during the census reference week. Because the focus is on paid employment, rather than the amount of time spent in the studio or the desire to sell art, many artists are likely to be overlooked—that is, not counted as artists. On an individual basis, this doesn’t matter as census information is anonymized, and no art dealers will throw artists out of their galleries because the Census Bureau didn’t classify them as artists. There is, however, a national policy downside: municipal, state and federal legislators are less likely to give money to the arts or to create laws that benefit artists if this group is significantly undercounted.

Artists devote time to their art . Although the National Endowment for the Arts makes use of Census Bureau data, the agency has conducted its own surveys of artists in the visual and performance spaces over the years. One of those surveys, “Visual Artists in Four Cities” (Houston, Minneapolis, San Francisco and Washington, D.C.), identified artists by their level of activity, i.e., how many hours per week they spend on art, counting those who had exhibited in some gallery or other art space in those cities over several years.

An artist turns a profit on their art . If the Census Bureau takes a broad, sweeping view of the definition of artist, the Internal Revenue Service takes a narrow one, examining individual taxpayers’ returns, and the federal agency has its own definition of artists as professionals. There are nine criteria that the IRS applies to separate professionals from hobbyists (an important distinction, as professionals may deduct their expenses, hobbyists may not).

• Is the activity carried on in a businesslike manner?
• Does the artist intend to make the artistic activity profitable?
• Does the individual depend in full or in part from income generated by the artistic work?
• Are business losses to be expected, or are they due to circumstances beyond the artist’s control?
• Are business plans changed to improve profitability?
• Does the artist have the knowledge to make the activity profitable?
• Has the artist been successful in previous professional activities?
• Does the activity generate a profit in some years and, if so, how much?
• Will the artist make a profit in the future?

The artist need not answer “yes” to every question to legitimately deduct business-related expenses – including art supplies and equipment, studio rental, travel (mileage, airfare, parking, tolls, meals and lodging), educational expenses (conferences, master classes, museum membership) and the cost of advertising and promotion (business cards, brochures, photography, postage and shipping), but the IRS demands proof that an artist has made a genuine effort to earn a profit in at least three years out of five.

Artistic credentials, which don’t usually matter to collectors, critics, dealers and curators, may also help an artist make a case that he or she is a professional for tax purposes. These include earning a bachelor’s or master’s degree in the fine arts, membership in an artists’ society, experience teaching art, inclusion in Who’s Who in American Art or a similar directory and an exhibition history.

An artist is someone who requires an artist’s studio . Several private and public agencies certify artists’ eligibility to rent or buy live-work loft space apartments that have both residential and studio components. Artist Certification committees are set up to evaluate applicants’ need for space and their qualifications as serious, but not necessarily professional, artists. According to the artist certification guidelines of the Boston Redevelopment Authority in Massachusetts, “Any artist who can demonstrate to a committee of peers that they have a recent body of work as an artist, and who requires loft-style space to support that work, is eligible.” The definition that these committees use is quite flexible, focusing on subjective factors (“…the nature of the commitment of the artist to his or her art form as his or her primary vocation rather than the amount of financial remuneration earned from his or her creative endeavor,” in the words of the Artist Certification Committee of New York City’s Department of Cultural Affairs) rather than a set of hard numbers like exhibitions dates, sales, awards, memberships or commissions, which would disqualify most applicants. There is no written definition of artist at Artspace, the Minnesota-based developer of live-work spaces for artists around the country, and ad hoc certification committees at various sites look at applicants’ work (“they’re not making qualitative judgments, though,” Artspace spokeswoman Sarah Parker told Observer) to gauge the individual’s reputation within the artistic community.

An artist is an “independent contractor.” Artists are generally self-employed, working in their own studios, setting their own hours and creating objects that are of their own design and making, but sometimes they do work for others. For instance, they may serve as a studio assistant, helping another artist with practical matters, or they may be commissioned to produce an artwork, such as a monument, portrait or mural. In these circumstances, the definition of artist matters a lot, principally because of the issue of copyright. Someone who is an employee is paid a salary (and, perhaps, receives benefits, such as health insurance, sick and vacation pay), works certain set hours per week and is given explicit instructions on what tasks to fulfill and how to fulfill them at the employer’s work site, and the output that the individual produces on the job, which could be anything from paintings to sculpture, belongs to the employer. It is not uncommon, however, for artists who hire assistants to get around the need to pay taxes for these employees by calling them independent contractors, potentially giving those assistants a legal basis to be considered joint authors of the artists’ work if they were involved directly with the finished pieces. That joint authorship would be dependent upon the degree to which the assistant could prove that his or her original ideas and decision-making are part of the final work, according to Joshua Kaufman, a Washington, D.C. lawyer who often represents artists.

There have been some lawsuits brought by former assistants against the artists for whom they worked for joint copyright ownership of works, but they have all been settled out of court. However, in 1989, Kaufman successfully argued before the United States Supreme Court the right of sculptor James Earl Reid to claim copyright ownership of a work that a nonprofit group had commissioned him to create.

An artist is someone whom funding agencies call an artist . Public agencies and private organizations also provide money for individual artists, but nary one has published a formal definition of what an artist is. “We put this into the hands of our panel members,” Julie Gordon Dalgleish , former program director for artist fellowships at the Bush Foundation, told Observer. “We exclude certain things,” such as straight journalism, from the literature category and instructional videos from the category of film and video, but “we accepted an application from someone who braids hair. We would look at a tattoo artist if we felt there was a strong vision, creative energy and perseverance.” By we, of course, she meant the panels that review artists’ applications for fellowships to decide who receives money. Panel members regularly debate the question of whether or not a craft artist is an artist—a topic with a decades-long history. More recent concerns involve new media and digital art in two dimensions and three dimensions.

Ultimately, for most purposes, artists are people who call themselves artists. According to Marcel Duchamp , the artist defines art, and it seems increasingly true that nowadays artists also define who and what they are. Definitions by nature are confining and restrictive, while art and its makers seek to be expansive and inclusive. It may be simpler to state what makes an artist a professional than what defines an artist. ‘Artist’ has become a universal label denoting creativity and using it is often a way to shine a light on someone who does something particularly well. Socially, artists are often defined by the positive (freedom-loving, convention-defying) or negative (egotistical, bohemian) characteristics that other people attribute to them. Part of an artist’s job is to understand how artists are seen and what is expected of them, whether that’s by a certification committee that wants to see their body of work, a funding source that wants to understand the artist’s proposal, a dealer who wants to see what they’re capable of or the government, which more often than not, just wants to see the receipts.

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The Assignment Problem

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The assignment problem can be understood as the issue how decision rights are allocated to the agents and organizational subunits of a society. In this wide interpretation, the assignment problem refers to three different, albeit related phenomena: i) the allocation of property rights to individuals, i.e. households, and to firms, ii) the allocation of decisions rights to the government (including its subunits) relative to the private sector, and, within the sphere of government, iii) the attribution of policy instruments to economic policy agents. Economic policy agents are the different layers of government, the central bank, and trade unions to name the most important ones. 1 Assigning decisions rights is not only a national problem but refers to international decision rules as well.

We can think of the assignment problem as an optimizing problem in which decision rights are allocated in such a way that a goal function is maximized. In this fundamental issue of institutional economics, the goal function is complex, including values such as individual freedom and equity as well as the economic criterion of efficiency. Phrasing the assignment problem in these terms, I am putting the problem as if we were to reinvent mankind’s institutional arrangements from scratch. This is, of course, only an abstract and a rather theoretical approach. In the real world, the allocation of property rights to agents is given, and the given allocation changes only marginally except for major upheavals as in the shift from communist central planning to the market economy. In the following we look for some principles for the allocation of decision rights.

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Siebert, H. (2001). The Assignment Problem. In: Berninghaus, S.K., Braulke, M. (eds) Beiträge zur Mikro- und zur Makroökonomik. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56606-6_35

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