assignment problem in transportation problem

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Assignment Problem

5.1  introduction.

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.

5.2  GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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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.

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Transportation Problem | Set 1 (Introduction)

• Transportation Problem | Set 6 (MODI Method - UV Method)
• Transportation Problem | Set 2 (NorthWest Corner Method)
• Transportation Problem | Set 4 (Vogel's Approximation Method)
• Transportation Problem Set 8 | Transshipment Model-1
• Transportation Problem | Set 5 ( Unbalanced )
• Transportation Problem | Set 3 (Least Cost Cell Method)
• Transportation Problem | Set 7 ( Degeneracy in Transportation Problem )
• Max Flow Problem Introduction
• Traveling Salesman Problem (TSP) Implementation
• Bitonic Travelling Salesman Problem
• Travelling Salesman Problem implementation using BackTracking
• Travelling Salesman Problem using Hungarian method
• Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
• Problem on Trains, Boat and streams
• History of Transportation
• Transportation in the United States
• Transportation and Economic Development
• Road Transport - Definition, Types, Examples
• Problem on Time Speed and Distance

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?

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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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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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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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Twins batter Mariners in 11-1 victory in series finale

Nolan o'hara | may 9, 2024.

The Twins had to face the best of the Seattle Mariners’ starting pitching during their four-game series at Target Field in Minneapolis. Turns out, that wasn’t a problem.

The Twins beat up on Mariners starter Logan Gilbert in an 11-1 victory Thursday afternoon at Target Field in Minneapolis in front of an announced crowd of 22,154. The Twins took three of the four games against the Mariners despite facing top-tier pitching throughout the series.

"Big-time series. It was just a big-time, collective offensive move that we made in the series," Twins manager Rocco Baldelli said. "... It might be the best starting pitching you're going to run into all year long. It's just guy after guy who has dominant-type stuff. And we went out there and scored runs the entire series. I don't know what more I could ask for from our offense and the way they did their jobs."

Gilbert entered Thursday’s game with a 1.69 earned-run averaged and hadn’t given up more than four runs and five hits in seven previous starts this season. 

The Twins (22-15) reset those highs to nine hits and eight runs, and they chased Gilbert after four innings, making it his shortest start of the season, too. 

Minnesota plated five runs in the first inning, getting three straight singles from Trevor Larnach, Max Kepler and Jose Miranda. Kepler’s single marked the 11th straight game he’s recorded a hit, the longest hitting streak of his career. It also plated the first run of the game, scoring Edouard Julien, who had led off the inning by drawing a walk.

Later in the inning with two outs and the bases loaded, Manny Margot hit a three-RBI double to left field that scored Larnach, Miranda and Kepler. Carlos Santana brought Margot home with an RBI single to center field to give the Twins an early and commanding 5-0 lead. 

"That was great," Margot said via a translator. "We know the opposing pitcher was one of the best in the league, so getting the lead in the first inning, it was going to be good for the team."

Margot went 1 for 4 with five RBIs.

The lead was certainly good for the team, and the Mariners (20-18) weren't able to do much offensively against Twins starter Pablo Lopez, who pitched 6 1/3 innings of one-run ball, allowing just four hits while fanning 10. It’s the second straight day a Twins starter finished with double-digit strikeouts as Chris Paddack fanned 10 in a 6-3 victory over the Mariners Wednesday night. 

Lopez picked up his fourth win of the season.

"Today was all about the offense," Lopez said. "Everyone knows how good pitching for the Seattle Mariners is, and they were matched up against a tough guy today. He had been dominating, so the way they were able to put at-bats (together) and make things happen, my mindset was just like, 'How can I do my job fast to bring the boys back in the dugout as soon as possible?'

"They made it fun. They made it fun to be a part of the game today, the series, just go out there and just try to do my job, just try to do my part, and they gave me confidence. They made my job easier, so I was trying to make it easier on them somehow."

Lopez certainly did that by delivering another stellar start. The lone run he gave up came in the second inning when Mitch Haniger led off the frame with a double and was later brought home on Luke Raley's single to right field a couple at-bats later.

But Ryan Jeffers, who's been on a tear, homered in the bottom of the frame to get the run right back.

That was the only run the Mariners would push across in Thursday's series finale.

The offense continued to deliver for the Twins all game long. In the fourth inning, Julien hit an RBI single that scored Santana, who led off the frame with a single of his own. Jeffers hit a sacrifice fly to left field that scored Austin Martin, who was walked earlier in the fourth.

Then in the fifth inning, the Twins added another run when Miranda doubled and was brought home by Margot, who grounded out to short. In the seventh inning, Kepler hit a 411-foot solo homer to right-center field off Mariners reliever Tyson Miller. Miranda later reached on an error and scored when Margot hit a grounder to third baseman Luis Urias, but Urias' throw to first was off the mark, allowing Margot to reach and take second base. Those seventh-inning runs made it an 11-1 game.

Josh Staumont, called up from Triple-A St. Paul on Wednesday, made his first appearance of the season in the ninth inning with a 10-run advantage. While he allowed one hit and walked one, Staumont also fanned three and closed out the game without allowing any damage.

NOLAN O'HARA

Transportation | BART equipment problem shuts down service…

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Breaking news, transportation | head-on crash in brentwood injures seven, transportation | bart equipment problem shuts down service between richmond and oakland, service didn’t resume until late morning; no bus bridge made available.

Service on BART’s orange line between the Richmond and Berryessa stations also was down intermittently.

The orange line began running again about 6 a.m., according to social media posts from BART. An update from the agency on its website at 8 a.m. said the orange line was again down.

The rest of BART’s system ran normally through the morning commute.

Crews troubleshot the problem, according to BART, but officials did not immediately say more about the specific issue.

During the delay, riders on the Richmond line were told they should seek other means to get to the MacArthur Station. The agency said that “while we do have extra staff at each station to help with buses, there isn’t a direct bus bridge.”

BART received aid from AC Transit, which ran buses from the Richmond station to its station at 19th and Broadway in Oakland — a ride that was expected to take about an hour and 25 minutes, as opposed to about 20 minutes on BART. AC Transit also was busing passengers from the El Cerrito del Norte, El Cerrito Plaza, North Berkeley, Downtown Berkeley and Ashby stations in between the stations and all the way to Oakland.

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Fisker is now facing a $13 million lawsuit, the latest challenge for the struggling electric vehicle startup.

Fisker — which recently warned employees of layoffs — has been accused of stiffing an engineering firm on payments over the development of its Pear and Alaska vehicles.

The lawsuit , filed in federal court this week, was first reported by TechCrunch .

In the suit, a Michigan-based subsidiary of Bertrandt says Fisker broke a development agreement signed in May 2022.

According to the complaint, Bertrandt agreed to help Fisker launch its Pear electric car by providing "engineering, design, and development services," but Fisker failed to pay for the work and put the agreement on an indefinite "pause."

In a statement to Business Insider, a Fisker spokesperson said the accusations are "without merit."

"It is a legally baseless and disappointing attempt by what has been a valued partner to extract from Fisker payments and intellectual property to which Bertrandt has no right to under the relevant agreements or otherwise," the Fisker spokesperson said.

The lawsuit also alleges that Fisker asked Bertrandt to help with its Alaska pick-up truck and agreed to pay a quote of $1.66 million for the work, although there was never a formal written agreement or payment for the work.

According to the lawsuit, the unpaid services total over $7 million. Bertrandt says in the suit that Fisker agreed in February 2024 to pay over $3.6 million toward the balance but didn't follow through.

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The suit also accuses Fisker of holding onto Bertrandt's intellectual property "related to the engineering and development of the PEAR and ALASKA vehicles." It seeks just under $13 million in damages.

Layoffs could be on the horizon

On Monday, Fisker sent an email to staff telling them they could be laid off in two months.

"Fisker is diligently pursuing all options to address our operating cash requirements, including maintaining discussions with prospective buyers and investors and exploring various restructuring alternatives," the company said in a memo to staff, according to three workers and an email seen by BI. "There is a possibility, however, that these efforts will not be successful."

The memo said the cuts would be immediate.

Last month, Fisker CEO Henrik Fisker told staff at an all-hands meeting that the startup was in talks with four automakers about a possible acquisition , according to a recording of the event seen by BI.

Fisker didn't say who the four companies were.

"They obviously need time to get to some diligence," he said.

Fisker told employees the company was working with Deutsche Bank to find a buyer.

Meanwhile, several sources told BI that Fisker has been using parts from its preproduction vehicles and its inventory of vehicles to fix some customers' cars.

One employee told BI that the parts were used in roughly 10 to 15% of fixes over the last few months.

A Fisker spokesperson denied those allegations earlier this week.

"No parts have been taken off these vehicles for use in customers cars," the spokesperson said. The spokesperson said parts may have been stripped off engineering vehicles "for analysis or to retrofit other engineering vehicles, but never customer vehicles."

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