the assignment problem econ

How to Solve the Assignment Problem: A Complete Guide

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Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

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

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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2 Linear Programming: Formulation and Graphical Method

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4 Transportation Problem

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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

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11 Introduction to game theory and its Applications

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Assignment Problems

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• Martin Beckmann 2  

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Suppose each member i from one class of objects (persons, firms) i  = 1, …, n is matched with one object j from another class of equal size (jobs, locations) j  = 1, …, n and the economic outcome is measurable in money terms a ij . Let x ij  = 1 when object i is assigned to object j and x ij  = 0 otherwise. The payoff of this matching is then \( {\displaystyle {\sum}_{ij}{a}_{ij}{x}_{ij}} \) . It represents gross profits (profits before wages or rents) in the assignment of persons to jobs and of firms to locations.

This chapter was originally published in The New Palgrave: A Dictionary of Economics , 1st edition, 1987. Edited by John Eatwell, Murray Milgate and Peter Newman

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Cost minimization and utility maximization, bibliography.

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Beckmann, M. (1987). Assignment Problems. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5_429-1

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Title: fair congested assignment problem.

Abstract: We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion. Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc... Congestion is anonymous (it only depends on the number n of agents in a given post). A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n. For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts. If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair. In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists. It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient. Our approach to ex ante fairness generalises to the model where each agent's congestion is weighted. Now the caps on posts depend only upon own weight and total congestion, not on the number of other agents contributing to it. Remarkably in both models these caps are feasible if and only if they give to each agent the right to veto all but (1/m) of their feasible allocations.

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A New Solution to the Random Assignment Problem

• Anna Bogomolnaia , H. Moulin
• Published in Journal of Economics Theory 1 October 2001
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A simple random assignment problem with a unique solution, the probabilistic serial mechanism with private endowments.

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A solution to the random assignment problem on the full preference domain

Random assignment under ordinal preferences: a separation characterisation, ordinal efficiency and implementing random assignment by sequential object lotteries, a new ex-ante efficiency criterion and implications for the probabilistic serial mechanism, random assignment with multi-unit demands, efficiency in random allocation with ordinal rules, the probabilistic serial and random priority mechanisms with minimum quotas.

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STRATEGY-PROOFNESS OF THE PROBABILISTIC SERIAL MECHANISM IN LARGE RANDOM ASSIGNMENT PROBLEMS

22 references, ordinal efficiency and dominated sets of assignments, ordinal efficiency and the polyhedral separating hyperplane theorem, scheduling with opting out: improving upon random priority, elicitation of honest preferences for the assignment of individuals to positions, the efficient allocation of individuals to positions, truncation strategies in matching markets-in search of advice for participants, on a conjecture by gale about one-sided matching problems.

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What's going on inside Iran as the world awaits retaliation against Israel?

TEHRAN — As the world waits for  Iran ’s next step abroad, the talk inside the Islamic Republic is not just of what a retaliatory strike against Israel would mean for the Middle East , but what it would mean for the country’s economy and freedoms.

NBC News spoke to several residents of Tehran as their leaders rejected Western calls to refrain from an attack that the United States has indicated could come this week. 

“I am not the kind of person who agrees with war or attacks at all,” said Omid, an IT specialist. “In this case, Iranian leaders should consider that publicly attacking Israel to showcase military power could have significant economic and political costs for Iran, especially given the current chaotic economic situation,” said Omid, who did not want his last name used out of fear of repercussions for speaking out. 

There is no doubt that if Iran were to attack, Israel would retaliate, he said, and no one can predict what might happen next. “If Iranian leaders are determined to pursue retaliation, they should do so covertly and without leaving evidence,” added Omid, 40.

The country’s newly installed president, Masoud Pezeshkian, was elected on a reformist platform last month and is most likely mulling the potential domestic ramifications of a spiraling conflict for a nation already reeling from years of crushing sanctions, as well as crackdowns on dissent after a wave of mass demonstrations in 2022 shook the foundations of the Islamic Republic. Hamas political leader Ismail Haniyeh, 62, was assassinated in the Iranian capital last month after attending Pezeshkian’s inauguration. Both Iran and Hamas have accused Israel of orchestrating the killing. Israel has not commented on the matter but is widely believed to have carried out the attack. 

Iran vowed “severe punishment” for the assassination, but there remains a lot of uncertainty about when and how it could act. 

President Joe Biden suggested Tuesday that Tehran may not retaliate at all if a cease-fire deal is reached to end the war in Gaza and free remaining hostages. A new round of talks is due to commence Thursday in Qatar. 

As the U.S. and its allies scrambled to try to ward off a broader conflict, Pezeshkian said during a call with British Prime Minister Keir Starmer that it was Iran’s “legal right” to respond. Foreign ministry spokesman Nasser Kanaani on Tuesday dismissed Western calls for Iran to avoid escalating tensions in the region.

Accusing Israel of having violated Iran’s “sovereignty and territorial integrity” by killing Haniyeh in Tehran, Kanaani said the calls for Iran to de-escalate “lack political logic” and “amount to open and practical support for the source of international crimes and terrorism in the region.”

But some in Tehran told NBC News that Iran could keep Israel on its toes without a major retaliation. 

“No attack needed,” Nima Hosseini, a government worker, said. “A response in kind and in due time is a better alternative. Of course keeping Israel on high alert via rhetoric and verbal threats can be a smart strategy this time around,” Hosseini, 50, said. 

Others sounded worried about the prospect of Iran being sucked into a wider conflict. “There is no one in the world who likes war, but I am very worried about it,” Taha Azimi, 40, a construction worker, said. “I believe that Iran will definitely respond to Israel’s attacks to regain its dignity.”

Haniyeh’s assassination is “unacceptable” for the leaders of Iran, he said, given that the killing of an important political partner in Tehran has caused many problems for them both inside and outside the country. 

“For example, people inside the country think that the government does not have the ability to control the security of the capital or people outside conclude that there are many spies in Iran’s security system,” Azimi said.

Iran launched a limited retaliatory attack against Israel in April , the first time Tehran had directly attacked the Jewish state. 

Delaram Nemati, 30, who works for an architecture firm, said Iran will carry out the same limited strike this time around, “without any danger.” Still, she is worried that even that could lead to a wide-scale conflict. “I worry about war but try not to think about it,” Nemati said. 

Amin Khodadadi reported from Tehran, and Yuliya Talmazan reported from London.

Yuliya Talmazan is a reporter for NBC News Digital, based in London.

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Price increases when demand exceeds supply are textbook economics. The question is whether, and how much, the pandemic yielded an excess take.

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Carbon Emissions from AI and Crypto Are Surging and Tax Policy Can Help

What do crypto assets and artificial intelligence have in common? Both are power hungry.

Because of the electricity used by high-powered equipment to “mine” crypto assets, one Bitcoin transaction requires roughly the same amount of electricity as the average person in Ghana or Pakistan consumes in three years. ChatGPT queries require 10 times more electricity than a Google search, due to the electricity consumed by AI data centers.

As the Chart of the Week shows, crypto mining and data centers together accounted for 2 percent of world electricity demand in 2022. And that share is likely to climb to 3.5 percent in three years, according to our estimates based on projections from the International Energy Agency . That would be equivalent to current consumption of Japan, the world’s fifth largest electricity user.

The climate impact of these activities—irrespective of their social and economic benefits—is cause for concern. A recent IMF working paper found that crypto mining could generate 0.7 percent of global carbon dioxide emissions by 2027. Extending the analysis to data centers (based on IEA estimates), means their carbon emissions could reach 450 million tons by 2027, or 1.2 percent of the world total.

The tax system is one way to steer companies toward curbing emissions. According to IMF estimates, a direct tax of $0.047 per kilowatt hour would drive the crypto mining industry to curb its emissions in line with global goals. If considering air pollution’s impact on local health as well, that tax rate would rise to $0.089, translating into an 85 percent increase in average electricity price for miners. Such a levy would raise annual government revenue of $5.2 billion globally and reduce annual emissions by 100 million tons (around Belgium’s current emissions).

For data centers, a targeted tax on their electricity use would need to be set at $0.032 per kilowatt hour, or $0.052 including air pollution costs. It is slightly lower than for crypto because data centers tend to be in locations with greener electricity. This could raise as much as $18 billion annually.

The situation today is the opposite: many data centers and crypto miners enjoy generous tax exemptions and incentives on income, consumption, and property. Considering the environmental damage, the lack of significant employment, and pressures on the electrical grid (possibly raising prices for households and reducing demand for the use of other low emissions goods, such as electric vehicles), the net benefits of these special tax regimes are unclear at best.

Policy incentives

On the flip side, AI applications could lead to smarter and more efficient power use, which some have posited could help ease electricity demand. The right policies can still incentivize developing AI applications with positive societal spillovers while addressing the environmental damage.

For policymakers, a broad carbon price coordinated across countries would be the best way to curb emissions, because it would encourage reduced fossil-fuel consumption, cleaner power sources, and improved energy efficiency. To limit global warming to 2 degrees, countries would need to introduce additional measures equivalent to a carbon price rising to $85 per ton by 2030 .

In the absence of a global carbon price, targeted measures could encourage crypto miners and data centers to use more energy-efficient equipment and may even motivate the adoption of less energy intensive crypto mining. Complementing electricity taxes with credits for zero-emission, bilateral power purchase agreements, and potentially renewable energy certificates would also help.

Cross-border coordination also remains important, as stricter measures in one location could encourage relocation to jurisdictions with lower standards.

As the window of opportunity for containing rising temperatures rapidly closes, expanding renewable energy sources and adopting an appropriate carbon price are urgently needed. In the interim, targeted measures, including taxation, can help mitigate increasing emissions from crypto mining and data centers.

Countries Can Tap Tax Potential to Finance Development Goals

Emerging market and low-income economies have a significant untapped tax potential of 8 percent to 9 percent of GDP

Harnessing GovTech to Tax Smarter and Spend Smarter

Digitalization, done right, can equip governments to improve revenue collection and spending efficiency.

Crypto Poses Significant Tax Problems—and They Could Get Worse

Tax systems need updating to cope with crypto assets, whose anonymity and decentralized nature poses challenges—not least for the value added tax.

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