assignment transportation problems

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.

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Transportation and Assignment Problems

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• 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.

• Assignment Problem
• Transportation Problem
• Basic Feasible Solution
• Unique Optimal Solution
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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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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?

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.

If you’re unable to see the whole table kindly convert the mobile view to the desktop view

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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Balanced and Unbalanced Transportation Problems

The two categories of transportation problems are balanced and unbalanced transportation problems . As we all know, a transportation problem is a type of Linear Programming Problem (LPP) in which items are carried from a set of sources to a set of destinations based on the supply and demand of the sources and destinations, with the goal of minimizing the total transportation cost. It is also known as the Hitchcock problem.

Introduction to Balanced and Unbalanced Transportation Problems

Balanced transportation problem.

The problem is considered to be a balanced transportation problem when both supplies and demands are equal.

Unbalanced Transportation Problem

Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.

Methods of Solving Transportation Problems

There are three ways for determining the initial basic feasible solution. They are

1. NorthWest Corner Cell Method.

2. Vogel’s Approximation Method (VAM).

3. Least Call Cell Method.

The following is the basic framework of the balanced transportation problem:

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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Guest Essay

The Problem With Saying ‘Sex Assigned at Birth’

By Alex Byrne and Carole K. Hooven

Mr. Byrne is a philosopher and the author of “Trouble With Gender: Sex Facts, Gender Fictions.” Ms. Hooven is an evolutionary biologist and the author of “T: The Story of Testosterone, the Hormone That Dominates and Divides Us.”

As you may have noticed, “sex” is out, and “sex assigned at birth” is in. Instead of asking for a person’s sex, some medical and camp forms these days ask for “sex assigned at birth” or “assigned sex” (often in addition to gender identity). The American Medical Association and the American Psychological Association endorse this terminology; its use has also exploded in academic articles. The Cleveland Clinic’s online glossary of diseases and conditions tells us that the “inability to achieve or maintain an erection” is a symptom of sexual dysfunction, not in “males,” but in “people assigned male at birth.”

This trend began around a decade ago, part of an increasing emphasis in society on emotional comfort and insulation from offense — what some have called “ safetyism .” “Sex” is now often seen as a biased or insensitive word because it may fail to reflect how people identify themselves. One reason for the adoption of “assigned sex,” therefore, is that it supplies respectful euphemisms, softening what to some nonbinary and transgender people, among others, can feel like a harsh biological reality. Saying that someone was “assigned female at birth” is taken to be an indirect and more polite way of communicating that the person is biologically female. The terminology can also function to signal solidarity with trans and nonbinary people, as well as convey the radical idea that our traditional understanding of sex is outdated.

The shift to “sex assigned at birth” may be well intentioned, but it is not progress. We are not against politeness or expressions of solidarity, but “sex assigned at birth” can confuse people and creates doubt about a biological fact when there shouldn’t be any. Nor is the phrase called for because our traditional understanding of sex needs correcting — it doesn’t.

This matters because sex matters. Sex is a fundamental biological feature with significant consequences for our species, so there are costs to encouraging misconceptions about it.

Sex matters for health, safety and social policy and interacts in complicated ways with culture. Women are nearly twice as likely as men to experience harmful side effects from drugs, a problem that may be ameliorated by reducing drug doses for females. Males, meanwhile, are more likely to die from Covid-19 and cancer, and commit the vast majority of homicides and sexual assaults . We aren’t suggesting that “assigned sex” will increase the death toll. However, terminology about important matters should be as clear as possible.

More generally, the interaction between sex and human culture is crucial to understanding psychological and physical differences between boys and girls, men and women. We cannot have such understanding unless we know what sex is, which means having the linguistic tools necessary to discuss it. The Associated Press cautions journalists that describing women as “female” may be objectionable because “it can be seen as emphasizing biology,” but sometimes biology is highly relevant. The heated debate about transgender women participating in female sports is an example ; whatever view one takes on the matter, biologically driven athletic differences between the sexes are real.

When influential organizations and individuals promote “sex assigned at birth,” they are encouraging a culture in which citizens can be shamed for using words like “sex,” “male” and “female” that are familiar to everyone in society, as well as necessary to discuss the implications of sex. This is not the usual kind of censoriousness, which discourages the public endorsement of certain opinions. It is more subtle, repressing the very vocabulary needed to discuss the opinions in the first place.

A proponent of the new language may object, arguing that sex is not being avoided, but merely addressed and described with greater empathy. The introduction of euphemisms to ease uncomfortable associations with old words happens all the time — for instance “plus sized” as a replacement for “overweight.” Admittedly, the effects may be short-lived , because euphemisms themselves often become offensive, and indeed “larger-bodied” is now often preferred to “plus sized.” But what’s the harm? No one gets confused, and the euphemisms allow us to express extra sensitivity. Some see “sex assigned at birth” in the same positive light: It’s a way of talking about sex that is gender-affirming and inclusive .

The problem is that “sex assigned at birth”— unlike “larger-bodied”— is very misleading. Saying that someone was “assigned female at birth” suggests that the person’s sex is at best a matter of educated guesswork. “Assigned” can connote arbitrariness — as in “assigned classroom seating” — and so “sex assigned at birth” can also suggest that there is no objective reality behind “male” and “female,” no biological categories to which the words refer.

Contrary to what we might assume, avoiding “sex” doesn’t serve the cause of inclusivity: not speaking plainly about males and females is patronizing. We sometimes sugarcoat the biological facts for children, but competent adults deserve straight talk. Nor are circumlocutions needed to secure personal protections and rights, including transgender rights. In the Supreme Court’s Bostock v. Clayton County decision in 2020, which outlawed workplace discrimination against gay and transgender people, Justice Neil Gorsuch used “sex,” not “sex assigned at birth.”

A more radical proponent of “assigned sex” will object that the very idea of sex as a biological fact is suspect. According to this view — associated with the French philosopher Michel Foucault and, more recently, the American philosopher Judith Butler — sex is somehow a cultural production, the result of labeling babies male or female. “Sex assigned at birth” should therefore be preferred over “sex,” not because it is more polite, but because it is more accurate.

This position tacitly assumes that humans are exempt from the natural order. If only! Alas, we are animals. Sexed organisms were present on Earth at least a billion years ago, and males and females would have been around even if humans had never evolved. Sex is not in any sense the result of linguistic ceremonies in the delivery room or other cultural practices. Lonesome George, the long-lived Galápagos giant tortoise , was male. He was not assigned male at birth — or rather, in George’s case, at hatching. A baby abandoned at birth may not have been assigned male or female by anyone, yet the baby still has a sex. Despite the confusion sown by some scholars, we can be confident that the sex binary is not a human invention.

Another downside of “assigned sex” is that it biases the conversation away from established biological facts and infuses it with a sociopolitical agenda, which only serves to intensify social and political divisions. We need shared language that can help us clearly state opinions and develop the best policies on medical, social and legal issues. That shared language is the starting point for mutual understanding and democratic deliberation, even if strong disagreement remains.

What can be done? The ascendance of “sex assigned at birth” is not an example of unhurried and organic linguistic change. As recently as 2012 The New York Times reported on the new fashion for gender-reveal parties, “during which expectant parents share the moment they discover their baby’s sex.” In the intervening decade, sex has gone from being “discovered” to “assigned” because so many authorities insisted on the new usage. In the face of organic change, resistance is usually futile. Fortunately, a trend that is imposed top-down is often easier to reverse.

Admittedly, no one individual, or even a small group, can turn the lumbering ship of English around. But if professional organizations change their style guides and glossaries, we can expect that their members will largely follow suit. And organizations in turn respond to lobbying from their members. Journalists, medical professionals, academics and others have the collective power to restore language that more faithfully reflects reality. We will have to wait for them to do that.

Meanwhile, we can each apply Strunk and White’s famous advice in “The Elements of Style” to “sex assigned at birth”: omit needless words.

Alex Byrne is a professor of philosophy at M.I.T. and the author of “Trouble With Gender: Sex Facts, Gender Fictions.” Carole K. Hooven is an evolutionary biologist, a nonresident senior fellow at the American Enterprise Institute, an associate in the Harvard psychology department, and the author of “T: The Story of Testosterone, the Hormone That Dominates and Divides Us.”

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

Follow The New York Times Opinion section on Facebook , Instagram , TikTok , WhatsApp , X and Threads .

Having trouble seeing through headlights while driving at night? You're not alone

Have you recently noticed more vision problems passing other cars while driving at night? It is likely not your imagination or your eyesight worsening with age.

Many Americans have reported similar issues, and factors such as larger vehicles and headlight misalignment are partially to blame. But the most significant aspect is a change in the headlights themselves causing eye discomfort and pain for other drivers.

Problems with night driving

In some sense, nighttime driving has always been difficult for some people. The reasoning has to do with a person's pupils, the round openings in the center of each eye, said Dr. Joseph Allen, an optometrist and host of the Doctor Eye Health YouTube channel with more than 1 million subscribers.

Pupils widen in the dark, a feature Allen compared to a "survival mechanism" that allows a person to detect movement more easily. The downside, he explained in one video , is it causes a loss of "depth of focus," meaning an object's edges will be slightly blurred even if the item itself is clear.

A sudden burst of headlights from an oncoming car can cause the pupils to rapidly constrict, leading to a brief but painful sensation. And while an unexpected flash of light is never particularly pleasant, modern cars are often equipped with lights that result in even more discomfort for drivers.

Older car headlights generally contained halogen lightbulbs that produced a yellowish color before switching to LED lights that give off white or blue hues in the past 20 years or so. According to Allen, research has shown that the newer headlight colors are more triggering to our neurological system even when the brightness between them and the halogen bulbs is the same.

"I think personally they should probably be made illegal in some way," the optometrist said of the LED lights. "When I see them, they not only are bothersome visually, like I can't see, but they cause almost a physical pain inside my eye."

Car size, alignment issues

Another key factor contributing to the headlight problem is the increasing size of cars. A Business Insider story , citing data from J.D. Power, said the number of new SUVs and trucks sold in the U.S. jumped from 53% in 2010 to 79% in 2021. The larger vehicles mean headlights are set higher, often shining directly in drivers' faces as opposed to older models set below eye level.

In addition, the headlights in many vehicles may not be aligned properly — and there are very few regulations in place to address it. According to an NBC news article from May 2023, only 10 states check headlight alignment as part of annual or biennial vehicle inspections.

As a result, many headlights are either tilted upward into the eyesight of other drivers or pointed in different directions. For the safety of others on the road, Allen recommends motorists check with a mechanic to ensure their car lights are properly aligned to point straight out and down.

Solutions and roadblocks

The technology to improve car headlights without blinding others already exists and is used in several countries. But the U.S. is not among them, and experts say it could be a while before that changes.

The beams, known as adaptive driving beam, or ADB, headlights, continuously adjust the amount of emanating light to shine less on occupied parts of the road and more on unoccupied parts. They have been used in Europe since 2012 and are sold today in every major auto market in the world besides the U.S., according to NBC.

Despite new safety regulations enacted in 2022, ADB technology is still not available for American cars. According to a CBS story from February, the various red tape laws and regulations make it difficult to add the feature, meaning it will likely be at least a few years until it becomes commonplace here.

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 from the slowdown.

The country's Passenger Car Association expects sales of new-energy vehicles to climb 25% to 11 million this year, according to figures reported by Bloomberg. 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 the Model 3, S, X, and Y in China last year in a bid to compete with local rivals including market leader BYD, which sells much cheaper vehicles such as the $11,000 Seagull .

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

Related stories

CEO Elon Musk appears to have backed away from the price cuts strategy this year, however. That's a sign he knows Tesla can't win the price war and remain profitable, Goldstein said.

"Last year was the year of the price cut, in order to grow volumes, and it worked," he said. "But Tesla now seems to have made the decision that for now, they're happy where their unit profits are at."

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

Bad news for BYD

Americans are shunning electric cars due to 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 as at the end of 2023, according to the China Electric Vehicle Charging Infrastructure Promotion Alliance. And the mid-market hatchback options offered by BYD 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 ($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 an ongoing property-market crisis fueling a decline in consumer spending.

BYD makes most of its sales in China — and the tally rose 43% to about 300,000 in the first three months of the year, according to a stock market filing this week. 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 and recently reduced production from six-and-a-half days a week to five, Bloomberg reported , 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: What happens when Elon Musk moves markets with a tweet

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