what is transportation problem and assignment problem

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Transportation Problem | Set 1 (Introduction)

Transportation problem is a special kind of Linear Programming Problem (LPP) in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the sources and destination respectively such that the total cost of transportation is minimized. It is also sometimes called as Hitchcock problem.

Types of Transportation problems: Balanced: When both supplies and demands are equal then the problem is said to be a balanced transportation problem.

Unbalanced: When the supply and demand are not equal then it is said to be an unbalanced transportation problem. In this type of problem, either a dummy row or a dummy column is added according to the requirement to make it a balanced problem. Then it can be solved similar to the balanced problem.

Methods to Solve: To find the initial basic feasible solution there are three methods:

• NorthWest Corner Cell Method.
• Least Cost Method.
• Vogel’s Approximation Method (VAM).

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Difference between transportation and assignment problems?

• Engineeringbro
• February 11, 2023
• March 10, 2024
• 3 mins read

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lets understand the Difference between transportation and assignment problems?

Transportation problems and assignment problems are two types of linear programming problems that arise in different applications.

The main difference between transportation and assignment problems is in the nature of the decision variables and the constraints.

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Additional Different between Transportation and Assignment Problems are as follows : 

Decision Variables:

In a transportation problem, the decision variables represent the flow of goods from sources to destinations. Each variable represents the quantity of goods transported from a source to a destination.

In contrast, in an assignment problem, the decision variables represent the assignment of agents to tasks. Each variable represents whether an agent is assigned to a particular task or not.

Constraints:

In a transportation problem, the constraints ensure that the supply from each source matches the demand at each destination and that the total flow of goods does not exceed the capacity of each source and destination.

In contrast, in an assignment problem, the constraints ensure that each task is assigned to exactly one agent and that each agent is assigned to at most one task.

Objective function:

The objective function in a transportation problem typically involves minimizing the total cost of transportation or maximizing the total profit of transportation.

In an assignment problem, the objective function typically involves minimizing the total cost or maximizing the total benefit of assigning agents to tasks.

In summary,

The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations,

while the assignment problem is concerned with finding the optimal way to assign agents to tasks.

Both problems are important in operations research and have numerous practical applications.

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Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

The whole model is:

subject to,

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

• Book:Operations Research

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• Introduction

Transportation Problem Explained and how to solve it?

• Transportation Problem
• Balanced Problem
• Unbalanced Problem

Contributed by: Patrick

Operations Research (OR) is a state of art approach used for problem-solving and decision making. OR helps any organization to achieve their best performance under the given constraints or circumstances. The prominent OR techniques are,

• Linear programming
• Goal programming
• Integer programming
• Dynamic programming
• Network programming

One of the problems the organizations face is the transportation problem. It originally means the problem of transporting/shipping the commodities from the industry to the destinations with the least possible cost while satisfying the supply and demand limits.  It is a special class of linear programming technique that was designed for models with linear objective and constraint functions. Their application can be extended to other areas of operation, including

• Scheduling and Time management
• Network optimization
• Inventory management
• Enterprise resource planning
• Process planning
• Routing optimization  

The notations of the representation are:

m sources and n destinations

(i , j) joining source (i) and destination (j)

c ij 🡪  transportation cost per unit

x ij 🡪  amount shipped

a i   🡪 the amount of supply at source (i)

b j   🡪 the amount of demand at destination (j)

Transportation problem works in a way of minimizing the cost function. Here, the cost function is the amount of money spent to the logistics provider for transporting the commodities from production or supplier place to the demand place. Many factors decide the cost of transport. It includes the distance between the two locations, the path followed, mode of transport, the number of units that are transported, the speed of transport, etc. So, the focus here is to transport the commodities with minimum transportation cost without any compromise in supply and demand. The transportation problem is an extension of linear programming technique because the transportation costs are formulated as a linear function to the supply capacity and demand. Check out the course on transportation analytics .

Transportation problem exists in two forms. 

• Balanced 

It is the case where the total supply equals the total demand.

It is the case where either the demand is greater than the supply, or vice versa.

In most cases, the problems take a balanced form. It is because usually, the production units work, taking the inventory and the demand into consideration. Overproduction increases the inventory cost whereas under production is challenged by the demand. Hence the trade-off should be carefully examined. Whereas, the unbalanced form exists in a situation where there is an unprecedented increase or decrease in demand.

Let us understand this in a much simpler way with the help of a basic example. 

Let us assume that there is a leading global automotive supplier company named JIM. JIM has it’s production plants in many countries and supplies products to all the top automotive makers in the world. For instance, let’s consider that there are three plants in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per day. The cost of transport per unit per km in INR and the distance between each source and destination in Kms are given in the tables below.

Here, the objective is to determine the unknown while satisfying all the supply and demand restrictions. The cost of shipping from a source to a destination is directly proportional to the number of units shipped.

Many sophisticated programming languages have evolved to solve OR problems in a much simpler and easier way. But the significance of Microsoft Excel cannot be compromised and devalued at any time. It also provides us with a greater understanding of the problem than others. Hence we will use Excel to solve the problem.

It is always better to formulate the working procedure in steps that it helps in better understanding and prevents from committing any error.

Steps to be followed to solve the problem:

• Create a transportation matrix (define decision variables)
• Define the objective function
• Formulate the constraints
• Solve using LP method 

Creating a transportation matrix:

A transportation matrix is a way of understanding the maximum possibilities the shipment can be done. It is also known as decision variables because these are the variables of interest that we will change to achieve the objective, that is, minimizing the cost function.

Define the objective function: 

An objective function is our target variable. It is the cost function, that is, the total cost incurred for transporting. It is known as an objective function because our interest here is to minimize the cost of transporting while satisfying all the supply and demand restrictions.

The objective function is the total cost. It is obtained by the sum product of the cost per unit per km and the decision variables (highlighted in red), as the total cost is directly proportional to the sum product of the number of units shipped and cost of transport per unit per Km.

The column “Total shipped” is the sum of the columns A, B, C, and D for respective rows and the row “Total Demand” is the sum of rows M, N, and O for the respective columns. These two columns are introduced to satisfy the constraints of the amount of supply and demand while solving the cost function. 

Formulate the constraints:

The constraints are formulated concerning the demand and supply for respective rows and columns. The importance of these constraints is to ensure they satisfy all the supply and demand restrictions.

For example, the fourth constraint, x ma + x na + x oa = 650 is used to ensure that the number of units coming from plants M, N, and O to customer A should not go below or above the demand that A has. Similarly the first constraint x ma + x mb + x mc + x md  = 700 will ensure that the capacity of the plant M will not go below or above the given capacity hence, the plant can be utilized to its fullest potential without compromising the inventory. 

Solve using LP method:

The simplest and most effective method to solve is using solver. The input parameters are fed as stated below and proceed to solve. 

This is the best-optimized cost function, and there is no possibility to achieve lesser cost than this having the same constraints.

From the solved solution, it is seen that plant M ships 100 units to customer A, 350 units to C and 250 units to D. But why nothing to customer B? And a similar trend can be seen for other plants as well. 

What could be the reason for this? Yes, you guessed it right! It is because some other plants ship at a profitable rate to a customer than others and as a result, you can find few plants supplying zero units to certain customers. 

So, when will these zero unit suppliers get profitable and can supply to those customers? Wait! Don’t panic. Excel has got away for it too. After proceeding to solve, there appears a dialogue box in which select the sensitivity report and click OK. You will get a wonderful sensitivity report which gives details of the opportunity cost or worthiness of the resource.

Basic explanation for the report variables,

Cell: The cell ID in the excel

Name: The supplier customer pairing

Final value: Number of units shipped (after solving)

Reduced cost: How much should the transportation cost per unit per km should be reduced to make the zero supplying plant profitable and start supplying

Objective coefficient: Current transportation cost per unit per Km for each supplier customer pair

Allowable Increase: It tells us the maximum cost of the current transportation cost per unit per Km can be increased which doesn’t make any changes to the solution

Allowable Decrease: It tells how much maximum the current transportation cost per unit per Km can be lowered which doesn’t make any changes to the solution

Here, look into the first row of the sensitivity report. Plant M supplies to customer A. Here, the transportation cost per unit per Km is ₹14 and 100 units are shipped to customer A. In this case, the transportation cost can increase a maximum of ₹6, and can lower to a maximum of ₹1. For any value within this range, there will not be any change in the final solution. 

Now, something interesting. Look at the second row. Between MB, there is not a single unit supplied to customer B from plant M. The current shipping cost is ₹22 and to make this pair profitable and start a business, the cost should come down by ₹6 per unit per Km. Whereas, there is no possibility of increasing the cost by even a rupee. If the shipping cost for this pair comes down to ₹16, we can expect a business to begin between them, and the final solution changes accordingly.

The above example is a balanced type problem where the supply equals the demand. In case of an unbalanced type, a dummy variable is added with either a supplier or a customer based on how the imbalance occurs.

Thus, the transportation problem in Excel not only solves the problem but also helps us to understand how the model works and what can be changed, and to what extent to modify the solution which in turn helps to determine the cost and an optimal supplier. 

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Table of contents

Transportation and Assignment Problems

Cite this chapter.

• James K. Strayer 2  

Part of the book series: Undergraduate Texts in Mathematics ((UTM))

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Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2–4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems. In the case of transportation problems, the algorithm is essentially a disguised form of the dual simplex algorithm of 4§2. Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems.

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Strayer, J.K. (1989). Transportation and Assignment Problems. In: Linear Programming and Its Applications. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1009-2_7

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

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WHAT IS TRANSPORTATION PROBLEM

The transportation problem is a special type of linear programming problem where the objective is to minimise the cost of distributing a product from a number of sources or origins to a number of destinations. Because of its special structure the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution. The origin of a transportation problem is the location from which shipments are despatched. The destination of a transportation problem is the location to which shipments are transported. The unit transportation cost is the cost of transporting one unit of the consignment from an origin to a destination.

In the most general form, a transportation ...

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Assignment Problem: Meaning, Methods and Variations | Operations Research

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

Meaning of Assignment Problem:

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

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

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

Definition of Assignment Problem:

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

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

The destinations D1, D2, D3, and D4 in the above table are where the products/goods will be transported from various sources O1, O2, O3, and O4. The supply from the source Oi is represented by S i . The demand for the destination Dj is d j . If a product is delivered from source Si to destination Dj, then the cost is called C ij .

Let us now explore the process of solving the balanced transportation problem using one of the ways known as the NorthWest Corner Method in this article.

Solving Balanced Transportation problem by Northwest Corner Method

Consider this scenario:

With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively. Demands for the destination D1, D2, D3, and D4 are 250, 350, 400, and 200, respectively.

The starting point for the North West Corner technique is (O1, D1), which is the table’s northwest corner. The cost of transportation is calculated for each value in the cell. As indicated in the diagram, compare the demand for column D1 with the supply from source O1 and assign a minimum of two to the cell (O1, D1).

Column D1’s demand has been met, hence the entire column will be canceled. The supply from the source O1 is still 300 – 250 = 50.

Analyze the northwest corner, i.e. (O1, D2), of the remaining table, excluding column D1, and assign the lowest among the supply for the appropriate column and rows. Because the supply from O1 is 50 and the demand for D2 is 350, allocate 50 to the cell (O1, D2).

Now, row O1 is canceled because the supply from row O1 has been completed. Hence, the demand for Column D2 has become 350 – 50 = 50.

The northwest corner cell in the remaining table is (O2, D2). The shortest supply from source O2 (400) and the demand for column D2 (300) is 300, thus putting 300 in the cell (O2, D2). Because the demand for column D2 has been met, the column can be deleted, and the remaining supply from source O2 is 400 – 300 = 100.

Again, find the northwest corner of the table, i.e. (O2, D3), and compare the O2 supply (i.e. 100) to the D2 demand (i.e. 400) and assign the smaller (i.e. 100) to the cell (O2, D2). Row O2 has been canceled because the supply from O2 has been completed. Column D3 has a leftover demand of 400 – 100 = 300.

Continuing in the same manner, the final cell values will be:

It should be observed that the demand for the relevant columns and rows is equal in the last remaining cell, which was cell (O3, D4). In this situation, the supply from O3 was 200, and the demand for D4 was 200, therefore this cell was assigned to it. Nothing was left for any row or column at the end.

To achieve the basic solution, multiply the allotted value by the respective cell value (i.e. the cost) and add them all together.

I.e., (250 × 3) + (50 × 1) + (300 × 6) + (100 × 5) + (300 × 3) + (200 × 2) = 4400.

Solving Unbalanced Transportation Problem

An unbalanced transportation problem is provided below. Because the sum of all the supplies, O1, O2, O3, and O4, does not equal the sum of all the demands, D1, D2, D3, D4, and D5, the situation is unbalanced.

The idea of a dummy row or dummy column will be applied in this type of scenario. Because the supply is more than the demand in this situation, a fake demand column will be inserted, with a demand of (total supply – total demand), i.e. 117 – 95 = 22, as seen in the image below. A fake supply row would have been introduced if demand was greater than supply.

Now this problem has been changed to a balanced transportation problem, and it can be addressed using any of the ways listed below to solve a balanced transportation problem, such as the northwest corner method mentioned earlier.

Frequently Asked Questions on Balanced and Unbalanced Transportation Problems

What is meant by balanced and unbalanced transportation problems.

The problem is referred to as a balanced transportation problem when both supplies and demands are equal. Unbalanced transportation is defined as a situation where supply and demand are not equal.

What is called a transportation problem?

The transportation problem is a type of Linear Programming Problem in which commodities are carried from a set of sources to a set of destinations while taking into account the supply and demand of the sources and destinations, respectively, in order to reduce the total cost of transportation.

What are the different methods to solve transportation problems?

The following are three approaches to solve the transportation issue:

• NorthWest Corner Cell Method.
• Least Call Cell Method.
• Vogel’s Approximation Method (VAM).

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China has a problem with electric cars, and that's bad news for Elon Musk

• Sales of electric cars in China are expected to rise at a slower pace this year.
• That's likely to be a problem for Tesla too.
• Elon Musk's company has also struggled to keep up with Chinese rivals' aggressive price cuts.

BYD and Tesla both posted anemic first-quarter sales this week, serving up a reminder that demand for electric cars appears to be stalling around the world.

China, the world's largest EV market, hasn't been immune to the slowdown.

The country's Passenger Car Association expects sales of new-energy vehicles to climb by 25%, to 11 million, this year, Bloomberg reported. That's a healthy rise, but still well below last year's growth rate of 36%.

Any signs of weakening demand in China are a red flag for Tesla, which is already struggling to keep up with its local rivals' aggressive price cuts.

"I think a big part of Tesla's first-quarter deliveries miss came from China," Seth Goldstein, an equities strategist for Morningstar who chairs the research firm's EV committee, told Business Insider. "There's a lot of price competition, and we're seeing consumers go to other brands with cheaper offerings."

Tesla waves the white flag

Tesla slashed the prices of its Models 3, S, X, and Y in China last year in a bid to compete with local rivals including the market leader, BYD, which sells much cheaper vehicles, such as the $11,000 Seagull .

The cuts helped Tesla log record delivery numbers and kept its share price high — but it still lost its title as the world's top EV seller to BYD in 2023.

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Holding prices steady in China appears to have backfired, though. Tesla missed Wall Street forecasts for deliveries — and Bloomberg estimated its market share in the world's second-largest economy had fallen to about 7% from about 11% in early 2023.

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Americans are shunning electric cars because of concerns about charging and the emergence of cheaper hybrids — but the reasons for the slowdown in China are more complex.

There are plenty of public charging stations — about 2.7 million at the end of 2023, according to the China Electric Vehicle Charging Infrastructure Promotion Alliance. And the midmarket hatchback options BYD offers are some of China's best-selling cars.

However, some would-be buyers found local companies' constant price cuts irritating . BYD slashed the cost of one model by a total of 15,000 yuan, or about $2,100, in a matter of months last year, reducing its value on the secondhand market and making people more hesitant to buy one.

China's economy has also struggled since the end of the pandemic, with deflationary pressures and a property-market crisis fueling a decline in consumer spending.

BYD makes most of its sales in China — and the tally rose by about 300,000 in the first three months of the year, a stock-market filing this week said. That increase was not enough to stop Tesla from reclaiming its title as the world's top electric-car maker despite its own dismal delivery numbers for the same period.

Tesla makes the Model 3 and Model Y at its Shanghai factory. Bloomberg reported that it recently reduced production to five days a week from 6 ½ days, in a sign of waning demand for its cars in China.

If China joins the US in loving EVs a little less, both Tesla and BYD could be in for a difficult 2024.

Watch: How did Tesla's bulletproof Cybertruck become so expensive and so delayed?

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Title: ita-ecbs: a bounded-suboptimal algorithm for combined target-assignment and path-finding problem.

Abstract: Multi-Agent Path Finding (MAPF), i.e., finding collision-free paths for multiple robots, plays a critical role in many applications. Sometimes, assigning a specific target to each agent also presents a challenge. The Combined Target-Assignment and Path-Finding (TAPF) problem, a variant of MAPF, requires simultaneously assigning targets to agents and planning collision-free paths. Several algorithms, including CBM, CBS-TA, and ITA-CBS, can optimally solve the TAPF problem, with ITA-CBS being the leading method of flowtime. However, the only existing suboptimal method ECBS-TA, is derived from CBS-TA rather than ITA-CBS, and adapting the optimal ITA-CBS method to its bounded-suboptimal variant is a challenge due to the variability of target assignment solutions in different search nodes. We introduce ITA-ECBS as the first bounded-suboptimal variant of ITA-CBS. ITA-ECBS employs focal search to enhance efficiency and determines target assignments based on a new lower bound matrix. We show that ITA-ECBS outperforms the baseline method ECBS-TA in 87.42% of 54,033 test cases.

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Ford to send 144,000 trucks to North American dealers

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Joe White is the global automotive correspondent for Reuters, based in Detroit. Joe joined Reuters in January 2015 as the transportation editor leading coverage of planes, trains and automobiles, and later became global automotive editor. Previously, he served as the global automotive editor of the Wall Street Journal, where he oversaw coverage of the auto industry and ran the Detroit bureau. Joe is co-author (with Paul Ingrassia) of Comeback: The Fall and Rise of the American Automobile Industry, and he and Paul shared the Pulitzer Prize for beat reporting in 1993. You can sign up for Joe's Auto File newsletter here: https://www.reuters.com/newsletters/reuters-auto-file/

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Ethiopia faces tough devaluation decision to secure IMF bailout

Ethiopia may have to decide on a big currency devaluation sooner rather than later to secure a rescue loan from the International Monetary Fund (IMF), which left the country last week without reaching a much-needed deal with authorities.

Falling parts and a ‘nosedive’: 6 probes involving Boeing planes this year

The engine cover on a Southwest Airlines Boeing 737-800 fell off during takeoff in Denver this week , striking a wing flap and forcing the plane to return immediately to the airport. It marked the latest in a slew of high-profile mishaps involving Boeing planes in 2024, ranging from lost wheels and engine failures to door plugs blowing out after takeoff. Nobody was seriously injured in any of those incidents.

Aviation regulators stress that flying is incredibly safe and fatal accidents exceedingly rare. Federal investigations into air incidents are relatively common and do not necessarily suggest that safety is at risk. The National Transportation Safety Board has opened more than 100 investigations in the United States this year so far, involving multiple aircraft manufacturers. The incidents also do not necessarily indicate issues were caused by the manufacturer, with some responsibilities belonging to the airlines instead.

Still, the incidents have increased scrutiny of Boeing’s safety record, quality control protocols and the airlines that operate its jets. In a February statement, Boeing said it had “taken important steps to foster a safety culture,” and that it will “continue our comprehensive efforts to improve our safety and quality programs.”

Here’s a roundup of incidents involving Boeing planes that have sparked investigations by aviation regulators this year. Boeing and the Federal Aviation Administration did not immediately respond to a request for comment for this article.

Door plug blows off fuselage on Alaska Airlines flight

On Jan. 5, the door plug on an Alaska Airlines Boeing 737 Max aircraft blew out while it was ascending over Portland, Ore., leaving a gaping hole in the side of the fuselage and forcing an emergency landing.

The door plug — an exit sealed with a panel rather than used as a door — was found in a teacher’s backyard. The seat closest to the breach was empty and none of the passengers were seriously injured, although some claimed physical injuries and emotional trauma in a lawsuit against Boeing.

The accident prompted intense media scrutiny and several investigations, including a criminal probe by the Justice Department. The FAA grounded 171 737 Max 9 planes over safety concerns. In February, a NTSB investigation found that the panel appeared to have been installed at a Boeing factory without four crucial bolts.

The FAA set a May 2024 deadline for Boeing to come up with a comprehensive plan to improve its quality control. Last week, Alaska said that Boeing had paid it $160 million in “initial compensation,” the Associated Press reported .

In a statement after the initial NTSB report, Boeing CEO Dave Calhoun said: “Whatever final conclusions are reached, Boeing is accountable for what happened. An event like this must not happen on an airplane that leaves our factory. We simply must do better for our customers and their passengers.”

Engine fails midflight on Atlas Air cargo carrier

One of the engines on a 747 cargo plane operated by Atlas Air failed on Jan. 18 after takeoff from Miami, forcing its pilot to make an emergency landing less than an hour after departing. A softball-sized hole was later found above its engine, and witness video appeared to show a plane flying partially aflame.

A preliminary NTSB report found that the aircraft sustained an “in-flight engine fire,” with the crew shutting down the engine and deploying the plane’s fire extinguishing system. The plane made an “uneventful landing” in Miami, where it was met by firefighting crew, the report said. There were no injuries, and the report said the aircraft sustained minor damage.

Boeing said at the time of the incident that it would support the NTSB’s investigation.

Delta flight loses wheel before takeoff

A Delta Air Lines Boeing 757 plane lost its nose tire while taxiing for takeoff in Atlanta on Jan. 23.

Passengers on the flight, destined for Bogotá, Colombia, were safely transferred to a replacement aircraft. According to Delta, the plane’s nose gear tire and rim had come loose. The FAA said the nose wheel rolled down a nearby hill and that it was investigating the incident.

At the time, a Boeing spokesperson said the 757, which the company stopped manufacturing in 2004, was 32 years old. Delta said it returned the aircraft to service the following day.

United plane loses tire during takeoff

A Japan-bound United Airlines lost a tire after takeoff on March 7, forcing the pilot to make an abrupt landing at Los Angeles International Airport.

Video appeared to show the plane taking off from San Francisco International Airport, before a tire comes loose from its left landing gear seconds later and falls to the ground.

According to United, the Boeing 777-200 aircraft has six tires on each of its two main landing gear struts and is designed to land safely with missing or damaged tires. The FAA said it opened an investigation.

Midair ‘nosedive’ during LATAM flight

A Boeing 787 flight from Sydney to Auckland, New Zealand, went into a dive on March 11, injuring some 50 people. The airline, LATAM, described the incident as a “technical event.”

Passengers described people getting thrown to the ceiling and then falling to the floor as the aircraft lost altitude — with one passenger saying he “felt the plane take a nosedive” — before being quickly leveled.

Chile’s national aviation opened a probe into the incident, with reports that investigators suspected one of the plane’s pilots had been pushed forward into the controls by his seat.

In a bulletin this year, reported by The Washington Post , Boeing reminded airlines of existing advice addressing an issue with pilot-seat switches. Were the switch to get stuck while someone was sitting in the seat, it could press their body against the plane’s controls, it said. The bulletin did not refer to the nosedive incident.

Engine cover falls off Southwest flight

On Sunday morning, the engine cover on a Southwest Airlines Boeing 737-800 fell off after takeoff, striking a wing flap. The Houston-bound flight landed safely in Denver after crew reported the issue.

Video obtained by ABC News and shared on social media shows the cover blowing open , then ripping off as the aircraft appears to move along the runway, exposing the engine to passengers.

A Boeing spokesperson referred inquiries to Southwest, which said in a statement that its maintenance teams were “reviewing the aircraft.”

Praveena Somasundaram, Ian Duncan, Justine McDaniel, Daniel Wu, Annabelle Timsit, Niha Masih and Kim Bellware contributed reporting.