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Assignment problem

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

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

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

(origins or supply),

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

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

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

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

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

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

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

This page was last edited on 5 April 2020, at 18:48.
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Original Paper
Published: 16 August 2021

The assignment problem revisited

Carlos A. Alfaro ORCID: orcid.org/0000-0001-9783-8587 1 ,
Sergio L. Perez 2 ,
Carlos E. Valencia 3 &
Marcos C. Vargas 1

Optimization Letters volume 16 , pages 1531–1548 ( 2022 ) Cite this article

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First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the $\epsilon $ - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.

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Acknowledgements

This research was partially supported by SNI and CONACyT.

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Banco de México, Mexico City, Mexico

Carlos A. Alfaro & Marcos C. Vargas

Mountain View, CA, 94043, USA

Sergio L. Perez

Departamento de Matemáticas, CINVESTAV del IPN, Apartado postal 14-740, 07000, Mexico City, Mexico

Carlos E. Valencia

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Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4

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DOI : https://doi.org/10.1007/s11590-021-01791-4

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

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WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

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An assignment problem is a special case of transportation problem where...

An assignment problem is a special case of transportation problem, where.

A. Number of rows equals number of columns

B. All rim conditions are 1

C. Values of each decision variable is either 0 or 1

D. All of the above

Answer: Option D

Solution(By Examveda Team)

This Question Belongs to Management >> Operations Research

Join The Discussion

Related Questions on Operations Research

The use of decision models.

A. Is possible when the variables value is known

B. Reduces the scope of judgement & intuition known with certainty in decision-making

C. Require the use of computer software

D. None of the above

Every mathematical model.

A. Must be deterministic

B. Requires computer aid for its solution

C. Represents data in numerical form

A physical model is example of.

A. An iconic model

B. An analogue model

C. A verbal model

D. A mathematical model

The qualitative approach to decision analysis relies on.

A. Experience

B. Judgement

C. Intuition

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The code that broke open the Murdaugh murder case was right at investigators’ fingertips. It still took months to crack.

Shattered collage showing Alex Murdaugh and family

A key piece of evidence in the murder trial of Alex Murdaugh was a cellphone video that captured his voice at the dog kennels on his family’s sprawling Lowcountry estate — the place where his wife, Margaret, and son Paul were found fatally shot on June 7, 2021.

The disgraced South Carolina lawyer had told authorities that he wasn't there the night of the killings. But the video — taken by Paul and stored on his phone — showed that he was.

It took investigators months to recover the video — even though the six-digit passcode to the locked device was as basic as it gets: 041499, Paul’s birthday.

In his first interview about the case, the South Carolina investigator charged with cracking the phone told NBC News' “Dateline” why the process took so frustratingly long, how law enforcement finally gained access to it, and how astonished authorities were when they discovered the video.

"I was in disbelief,” Lt. Britt Dove, a computer crimes investigator for the state Law Enforcement Division. “I hollered out that I found it to nobody in particular ’cause I was in the office working by myself.”

Murdaugh, who has proclaimed his innocence in the killings of Margaret, 52, and Paul, 22, has acknowledged lying to authorities about his alibi on the night of the murders.

Alex Murdaugh stands during his trial for murder at the Colleton County Courthouse

During his trial, Murdaugh, 55, blamed the deception on his addiction to pain pills and paranoia. He was convicted in March and sentenced to life in prison without parole.

Lawyers for Murdaugh have appealed the conviction, alleging in court documents filed in September that a county official tampered with the jury. An appeals court agreed to allow Murdaugh, who pleaded guilty Friday in a separate financial crimes case , to ask for a new trial.

Dove said he got Paul’s iPhone on Aug. 13, 2021, nine and a half weeks after the slayings. Investigators already knew that accessing it would be critical: The night of the killings, Alex told the case’s lead detective, David Owen, a senior special agent with the South Carolina Law Enforcement Division, that he hadn’t been to the kennels.

Murdaugh told Owen that he’d stayed at a house on the property and that after dinner, he’d taken a nap while his wife went to the kennels, Owen told “Dateline” in his first interview on the case.

At 10:07 p.m., Alex dialed 911 and said that he’d found the bodies of his wife and son.

But within 24 hours, Owen said, a Murdaugh family friend had provided investigators with details challenging that account. Paul was caring for the friend’s dog — a chocolate lab named Cash — and he called the friend on the night of the killings to talk through an issue he was having with Cash.

The friend, Rogan Gibson, told investigators that while talking with Paul, he was nearly certain that he heard Alex in the background, Owen said. Gibson mentioned something else: Paul had tried to send him a video of the dog but couldn't because of spotty cell coverage, Owen said.

The passcode problem

After the killings, authorities obtained Paul’s iPhone almost immediately, Owen said. But it was locked and authorities didn’t know his passcode.

Apple allows users to enter the wrong passcode only a handful of times before the phone is permanently disabled. To access the device after that, it needs to be reset — a process that would wipe out everything on it, including videos.

When Dove got the phone, he said it was temporarily disabled, with a message telling him to enter a passcode again in a few minutes. He said he wasn’t sure who had entered the wrong code or how many times they’d done so.

Investigators had supplied Dove with roughly 20 numbers that he described as significant dates associated with Paul, he said. Some were birthdays, Dove said, but none were Paul’s.

Although a date of birth might appear to be an obvious choice for a passcode, Dove said that in his experience — he’s been a computer crimes investigator for 15 years — the codes that people use vary.

Some select random numbers, Dove said. Others use street numbers or a spouse’s birthday.

"You can’t say the top three will always be a birthday,” he said. “It just changes from person to person.”

Dove tried a couple of combinations from the list, but they didn’t work, he said. And he tried breaking in with a “brute force attack,” or a forensic tool that systematically tries different combinations on a phone until it comes up with the correct one.

Those tools, which allow investigators to manually sidestep entering a passcode, can take years depending on the code’s complexity, Dove said. In a separate case, Dove recalled, it took authorities two years to break into a phone using a brute force attack.

More 'Dateline' cases

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After a limo driver disappears, a fugitive’s web begins to unravel and sets off an international manhunt
A killer known for his ‘zombie hunter’ costume is on death row, but some suspect he has another victim
His wife was killed the day before their eviction . A decade later, he faced another foreclosure — and a murder charge.
Illinois nurse who was terrorized by her ex turned to the court for help. She was dead a year later.
Days before her death, a Wisconsin woman sent an ominous letter: ‘He would be my first suspect ’

The technique recovered little on Paul's phone, and investigators were still unable to search the entire device, Dove said. So he decided to hold off and wait for a technological advance that might help them crack Paul’s passcode.

"I had to be mindful that if we try these attempts and we failed that we could have potentially lost any information on there,” he said. “And once we lost that information — even six months from now, a year from now, [when] we might have been able to get into it — it does us no good because the information’s gone.”

As the months ticked by and the investigation stalled, Dove said he eventually learned of a company that could likely access the phone, figure out its passcode and process the device. The U.S. Secret Service would assist in the process, Dove said, so he sent the phone to the agency.

But before federal investigators shared the device, Dove said, they decided to make one more attempt to unlock the phone using a passcode that Dove hadn’t tried — Paul’s birthday.

It worked, Dove said, and by late March 2022, he’d retrieved the device and obtained a copy of the video described by Gibson.

The crucial clip

The brief clip, which had a time stamp off 8:44 p.m., contained three distinct voices, Dove said — Paul, who’s shooting the video and grabbing at the dog, Cash; his mother, who can be heard shouting that another dog, Bubba, has a “bird” in its mouth; and a third voice — Alex’s — that was also shouting at Bubba.

Dove said he watched the video once, then turned up the volume and watched it again.

“Then I put headphones on to make sure I was really hearing what I thought I heard,” he said, adding: “This, we knew for a fact then, just destroyed whatever alibi that was put out.”

After watching the video a few times, he called Owen, who recalled thinking that he now had “tangible evidence” that Alex was lying.

"It put Alex at the kennels when he said he wasn’t there,” Owen said, adding: “I was excited. I was really excited.”

Alex’s lawyers came to have a name for what the video showed — “the lie.”

"How do you get around it?” Dick Harpootlian, one of those lawyers, told “Dateline.” “How do you explain it?”

"Apparently,” Harpootlian said of his client, “he didn’t.”

Tim Stelloh is a breaking news reporter for NBC News Digital.

Haylee Barber is a producer for NBC News' "Dateline."

Carol Gable is a producer for "Dateline."

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The election interference case against Trump is taking shape

Carrie Johnson

Then-President Donald Trump gestures as he arrives to speak at a rally in Washington, D.C., on Jan. 6, 2021. Jacquelyn Martin/AP hide caption

Then-President Donald Trump gestures as he arrives to speak at a rally in Washington, D.C., on Jan. 6, 2021.

The federal election interference case against former President Donald Trump is coming into sharper focus, as prosecutors assert he is responsible for the violence at the Capitol on Jan. 6, 2021, and offer new clues about how they intend to prove it.

The case set for trial in Washington, D.C., in March accuses Trump of leading a conspiracy to obstruct the certification of the 2020 election and deprive millions of voters from having their ballots count. More than 140 law enforcement officers suffered injuries after a mob stormed the U.S. Capitol on Jan. 6, disrupting the peaceful transfer of power.

Lawyers for Trump asked the judge to remove mentions of what happened that day from the four-count felony indictment because they could "inflame" or prejudice the jury.

"[N]ot a shred of evidence suggests President Trump called for any violence or asked anyone to enter the Capitol unlawfully," Trump attorneys John Lauro and Todd Blanche wrote in a court filing this week.

Trump is arraigned and pleads not guilty to charges he conspired to overturn election

Prosecutors say Trump should not be able to distance himself from the mayhem. They say evidence of the attack on the Capitol is "powerful and probative" about his motive and intent.

"Indeed, that day was the culmination of the defendant's criminal conspiracies to overturn the legitimate results of the presidential election, when the defendant directed a large and angry crowd — one that he had summoned to Washington, D.C., and fueled with knowingly false claims of election fraud — to the Capitol," wrote senior assistant special counsels Molly Gaston and Thomas Windom.

On Friday afternoon, U.S. District Judge Tanya Chutkan rejected Trump's bid to remove "inflammatory" allegations about Jan. 6 from the indictment. She said she never planned to show the jury a copy of the indictment, and that the jury selection process early next year should address how pretrial publicity, "including any generated by Defendant, has had on the impartiality of potential jurors."

The back and forth over how the events of Jan. 6 will play out in the case represents an early skirmish in one of the most important and closely watched prosecutions in American history.

For the first time, the Justice Department team said they are going to use video evidence to show Trump encouraged the crowd at the rally to go to the Capitol, starting 15 minutes into his speech. They said they will provide testimony, photos and geolocation evidence — essentially, cell phone pings — to show how specific Trump supporters listened to him, then went on to strong-arm police and breach the Capitol.

Read Trump's Jan. 6 Speech, A Key Part Of Impeachment Trial

Read Trump's Jan. 6 speech at the rally before the Capitol riot

Prosecutors said they plan to use other testimony and videos to demonstrate how Trump deployed angry rioters as a "tool" in his pressure campaign against then-Vice President Mike Pence.

"I think that one of the most material things that he did during that time was his 2:24 p.m. tweet about Mike Pence, essentially arguing to all of his followers that Mike Pence did not have the courage to do what needed to be done and that was while he knew that the Capitol was under attack," said Georgetown University Law Center professor Mary McCord, who is closely following the case.

Members of the mob ultimately chanted "hang Mike Pence" and "traitor Pence" as he hid in fear for his life alongside family members and Secret Service agents on Capitol grounds. Trump watched the events on television and remained "indifferent" to Pence's evacuation, prosecutors said.

McCord, who once led the Justice Department's National Security Division, said it's likely Trump will file more court papers as the trial looms to try to bar prosecutors from introducing specific pieces of evidence from Jan. 6.

Indeed, Lauro and Blanche signaled in a footnote this week they would seek to prevent jurors from hearing about acts of "extraordinary violent and destructive" behavior. The final decisions on evidentiary issues will be up to Judge Chutkan .

Trump's lawyers said the former president called for the rally crowd to be "peaceful and patriotic." They asserted that most of the crowd at the rally on Jan. 6 did behave peacefully.

But in their own filing, prosecutors said Trump continued to embrace members of the mob, including rioters charged with violence against law enforcement and detained pending trial in the D.C. jail. Trump has said on the campaign trail that he is considering pardons for people convicted of crimes related to Jan. 6.

"The defendant's decision to rally behind January 6 rioters and their cause is relevant to the jury's determination of whether he intended the actions at the Capitol that day," Gaston and Windom wrote.

U.S. Capitol insurrection
president donald trump
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Using the case study, assignment instructions, Content readings,...

Using the case study, assignment instructions, Content readings, and external research, develop your Section III. Requirements. The case study tells you that the executives and employees at Maryland Technology Consultants (MTC) have identified a need for an effective and efficient applicant tracking or hiring system. As you review the case study, use the assignment instructions to take notes to assist in your analysis. In particular, look for information in the interviews to provide stakeholder interests and needs.

Use the outline format, headings and tables provided and follow all formatting instructions below.

III. Requirements

Stakeholder Interests - Review the interest or objectives for the new hiring system for each stakeholder listed below based on his or her organizational role and case study information. Consider how the technology will improve how his/her job is done; that is, identify what each of the stakeholders needs the hiring system to do. Then to complete the table below, use information from the stakeholder interviews and identify one significant challenge or problem for each stakeholder related to the current hiring process (not their future expectations) . Then explain how a system could address their problems. Do not define what that position does in the organization. (Provide an introductory sentence for this section, copy the table below and complete the two columns with 1-2 complete sentences for each role in each column.)

B. Defining Requirements - The next step is to identify the essential requirements for the information system. In addition to the stakeholder interests identified above, review the Case Study, especially the interviews, highlighting any statements that tell what the person expects or needs the system to do. User requirements express specifically what the user needs the system to do . This can be in terms of tasks the users need to perform, data they need to input, what the system might do with that data input, and output required. System performance requirements express how the system will perform in several performance areas and security . As a member of the CIO's organization, you will use your professional knowledge to Identify 5 User Requirements (including one specifically related to reporting) and 5 System Performance Requirements (including 2 security-related requirements). Refer to Week 5 content on requirements; security requirements are covered in Week 6. Additional research can expand your knowledge of these areas.

Once you have identified the 10 requirements, evaluate each one using the criteria below and create 10 well-written requirements statements for the new hiring system.

The requirement statement:

Is a complete sentence, with a subject (system) and predicate (intended result, action or condition).
Identifies only one requirement; does not include the words "and," "also," "with," and "or."
For User Requirements, states what tasks the system will support or perform.
For System Performance Requirements, states how the system will perform.
Includes a measure or metric that can be used to determine whether the requirement is met (time or quantity), where appropriate.
Is stated in positive terms and uses "must" (not "shall," "may" or "should"); "the system must xxxx" not "the system must not xxx".
Avoids the use of terms that cannot be defined and measured, such as "approximately," "robust," "user friendly," etc.
Is achievable and realistic; avoids terms such as "100% uptime," or "no failures".

For a full requirements document, there will be many requirement statements; you only need to provide the number of requirements identified for each category. Do not provide generic statements but relate to the needs of MTC to improve its hiring process.

(Provide an introductory sentence, copy the table, and complete the Requirements Statement and Stakeholder columns. No additional information should be entered into the first column, Requirement ID.)

Formatting Your Assignment

Consider your audience - you are writing in the role of an MTC business analyst and your audience is MTC and your boss, the CIO. Don't discuss MTC as if the reader has no knowledge of the organization. Use third person consistently throughout the report. In third person, the writer avoids the pronouns I, we, my, and ours. The third person is used to make the writing more objective by taking the individual, the "self," out of the writing. This method is very helpful for effective business writing, a form in which facts, not opinion, drive the tone of the text. Writing in the third person allows the writer to come across as unbiased and thus more informed.

In Stage 3, you are preparing the third part of a 4-stage report. Use the structure, headings, and outline format provided here for your report. Use the numbering/lettering in the assignment instructions as shown below.